In the following, I would like to discuss a bit about the relation between finiteness in projective module classification and finite generation for K-theory. In particular, I would like to discuss why the affine surface in Daniel Litt's comment should not give rise to a counterexample.

  I would like to restrict to the case of low-dimensional smooth affine varieties over finite fields. Note that the following is not a precise argument and should only convey philosophical intuition: I am using some boundary cases of results from $\mathbb{A}^1$-homotopy theory (such as finite base fields...) and there would be considerably more work actually turning this into a proof.

The first comment concerns the UFD requirement. The finite generation for the Picard group is known. This implies a positive answer to the Bass conjecture for $K_0$ in case of curves over finite fields. Using that the Picard group is finitely generated, I think you can replace the UFD requirement by classifying projective modules up to isomorphism and tensoring by line bundles.

Second, the weak Bass conjecture implies the Bass conjecture: by a result of Serre (Theorem 1 in J.-P. Serre: Modules projectfs et espaces fibrés à fibre vectorielle. Séminaire Dubreil, Dubreil-Jacotin et Pisot, 1957/1958, Fasc 2, Exposé 23, p. 18, Sécretariat mathématique, Paris, 1958), if $X$ is an affine scheme of Krull dimension $d$, then all vector bundles of rank $>d$ split off a free direct summand. But this means that there is a global upper bound for the number of projective modules of given rank $n$, namely the sum of the numbers of projective modules of rank $\leq d$. The weak Bass conjecture would say that this is a finite number. But then, the group completion of the monoid of isomorphism classes of projective modules will be finitely generated, so we have the Bass conjecture.

Now, as a third point, I would like to discuss the $\mathbb{A}^1$-homotopy approach to the classification of projective modules, and outline how one might use that to understand the relation between the Bass conjecture and the finiteness of projective modules. In particular, the arguments below should convince you that the weak Bass conjecture holds for smooth affine varieties of dimension at most $3$, provided a refined version of Bass' conjecture and the Parshin conjecture hold.

Fabien Morel (in his book on $\mathbb{A}^1$-algebraic topology) has proved that for $X$ a smooth affine variety, there is a bijection between rank $n$ projective modules over $X$ and $\mathbb{A}^1$-homotopy classes of maps $X\to BGL_n$. Note in Morel's book there is a restriction $n\neq 2$ but that can be lifted using work of his student Moser. With this, it is possible to use classical topology obstruction methods to classify projective modules over smooth algebras. There is a recent series of amazing papers by Aravind Asok and Jean Fasel in which they are pursuing this line of thought. I would in particular like to point to arXiv:1204.0770, where they show that for a smooth affine $3$-fold over an algebraically closed field of characteristic $\neq 2$, the rank $2$ vector bundles can be classified by their Chern classes.

Their main theorem is for algebraically closed fields, but their methods adapt to finite fields (philosophically, but as mentioned above this would need quite some checking). Provided all that checking can be done, here is how the classification of rank two projective modules over smooth $\leq 3$-folds over finite fields would proceed (glossing over a good deal of details, sorry):
let $X$ be a smooth affine variety of dimension $\leq 3$. The result uses the Postnikov tower for $BGL_2$, and the classification of projective modules then can be made precise in terms of lifting classes in group $H^i_{\operatorname{Nis}}(X,\pi_i(BGL_2))$. To show ``finite generation'' of the corresponding set of isomorphism classes of projective modules, one needs finite generation results for such Nisnevich cohomology groups. The relevant homotopy group sheaves for $BGL_2$ are $\pi_1^{\mathbb{A}^1}BGL_2\cong \mathbb{G}_m$ (Morel-Voevodsky), $\pi_2^{\mathbb{A}^1}BGL_2\cong K^M_2$ (Morel) and $\pi_3^{\mathbb{A}^1}BGL_2$ is an extension of $KSp_3$ by some quotient of Milnor $K_4$ and the fifth power of the fundamental ideal in the Witt ring (Asok-Fasel, in the paper cited above). Now the relevant Nisnevich cohomology groups are $H^1(X,\mathbb{G}_m)$, $H^2(X,K^M_2)\cong CH^2(X)$ and $H^3(X,KSp_3)$ is the cokernel of $Sq^2:Ch^2(X)\to Ch^3(X)$. Over a finite field, we see that we need finite generation for $Pic(X)$, $CH^2(X)$ and $Ch^3(X)$ to show that there is only some finitely generated amount of projective modules. Note that in Daniel Litt's example, the rank two vector bundles would be classified exactly by their second Chern class in $CH^2(X)$.

Now what do the above groups have to do with the Bass conjecture? The relation is not that direct, Chow groups are not K-theory. Nevertheless, the conjecture is that Chow groups are finitely generated - this is called ``refined Bass conjecture'' in these talk notes of Thomas Geisser. In those same talk notes, you can also find Parshin's conjecture, which would imply that $CH^2(X)$ and $CH^3(X)$ are not just finitely generated but also torsion (if the base field is finite). Therefore, finiteness of rank two vector bundles in Daniel Litt's example would follow from finite generation for $CH^2(X)$ and Parshin's conjecture.

There is a disclaimer here: I would not expect that the classification of projective modules via $\mathbb{A}^1$-homotopy can be done for arbitrary dimensions. Nevertheless, one expects that there is a range (called metastable range by Suslin) in which the projective module classification can be done in terms of Chow groups, K-theory, hermitian K-theory and operations connecting them. In this range, finiteness of projective modules (over function rings of smooth affine schemes) would follow from finite generation of Chow groups (refined Bass conjecture) and Parshin's conjecture. In the end, the question does not seem a weak Bass conjecture to me - at least the way sketched above needs a lot more finite generation results as input, and we have not started talking about the situation of finitely generated $\mathbb{Z}$-algebras yet...

The following two statements might clarify the relation between Bass conjecture (for $K_0$) and its weak version in the question. I only consider the case of commutative rings.

1. Contrary to the formulation of the question, the weak Bass conjecture implies the Bass conjecture for $A$ finitely generated over $\mathbb{F}_q$.
2. The weak version of the Bass conjecture for $A$ smooth of dimension $\leq 3$ over a finite field can be deduced (using $\mathbb{A}^1$-homotopy classification of projective modules) from a refined version of the Bass conjecture and the Parshin conjecture.

I give a more detailed sketch of arguments below:

Ad 1: Serre proved that if $X$ is an affine scheme of Krull dimension $d$, then all vector bundles of rank $>d$ split off a free direct summand, cf. Theorem 1 in J.-P. Serre: Modules projectifs et espaces fibrés à fibre vectorielle. Séminaire Dubreil, Dubreil-Jacotin et Pisot, 1957/1958, Fasc 2, Exposé 23, p. 18, Sécretariat mathématique, Paris, 1958. This result implies that there is a global upper bound for the number of projective modules of given rank $n$ on an affine scheme of dimension $d$, namely the sum over $1\leq i\leq d$ of the numbers of projective modules of rank $i$. The weak Bass conjecture would say that this is a finite number. But then, the group completion of the monoid of isomorphism classes of projective modules will be finitely generated, so we have the ($K_0$-case of the) Bass conjecture. So the weak version of Bass' conjecture is in fact a strong version of the Bass conjecture.

The result of Serre also implies that if the weak version of the Bass conjecture holds for an affine variety of dimension $d$ over $\mathbb{F}_q$ and all ranks $\leq d$, then it holds for all ranks.

Ad 2: Consider $A$ a smooth variety of dimension $\leq 3$ over $\mathbb{F}_q$. The following is a sketch of an argument that the refined Bass conjecture and the Parshin conjecture imply the weak version of the Bass conjecture in the question. This is not entirely precise, as it uses some boundary cases of results from $\mathbb{A}^1$-homotopy theory (such as finite base fields...) and there might be considerable work turning this into a proof. I still think it conveys a good intuition for the strength of the "weak Bass conjecture". The basic idea is that projective modules over a smooth algebra of dimension $\leq 3$ are classified essentially by their Chern classes which lie in Chow groups $CH^i(X)$, the refined Bass conjecture and Parshin conjecture together imply that these Chow groups are finite.

First of all, the homotopy classification of projective modules: Fabien Morel (in his book on $\mathbb{A}^1$-algebraic topology) has proved that for $X$ a smooth affine variety, there is a bijection between rank $n$ projective modules over $X$ and $\mathbb{A}^1$-homotopy classes of maps $X\to BGL_n$. Note in Morel's book there is a restriction $n\neq 2$ but that can be lifted using work of his student Moser. With this, it is possible to use classical topology obstruction methods to classify projective modules over smooth algebras. There is a recent series of amazing papers by Aravind Asok and Jean Fasel in which they are pursuing this line of thought. I would in particular like to point to arXiv:1204.0770, where they show that for a smooth affine $3$-fold over an algebraically closed field of characteristic $\neq 2$, the rank $2$ vector bundles can be classified by their Chern classes. Although this result needs algebraically closed fields, the methods adapt to finite fields: the proof uses the Postnikov tower for $BGL_2$, and the classification of projective modules then can be made precise in terms of lifting classes in group $H^i_{\operatorname{Nis}}(X,\pi_i(BGL_2))$. 

The lifting groups and projective module classification: the relevant homotopy group sheaves for $BGL_2$ for classification of projective modules over varieties of dimension $\leq 3$ are $\pi_1^{\mathbb{A}^1}BGL_2\cong \mathbb{G}_m$ (Morel-Voevodsky), $\pi_2^{\mathbb{A}^1}BGL_2\cong K^M_2$ (Morel) and $\pi_3^{\mathbb{A}^1}BGL_2$ is an extension of $KSp_3$ by some quotient of Milnor $K_4$ and the fifth power of the fundamental ideal in the Witt ring (Asok-Fasel, in the paper cited above). Now the relevant Nisnevich cohomology groups are $H^1(X,\mathbb{G}_m)$, $H^2(X,K^M_2)\cong CH^2(X)$ and $H^3(X,KSp_3)$ is the cokernel of $Sq^2:Ch^2(X)\to Ch^3(X)$. This is not entirely honest, there is some finite amount from orientability issues because $\mathbb{F}_q$ is not quadratically closed. Anyway, over a finite field, the major information for projective module classification lies in $Pic(X)$, $CH^2(X)$ and $Ch^3(X)$.

Finiteness conjectures: Now what do the above groups have to do with the Bass conjecture? The relation is not that direct, Chow groups are not K-theory. Nevertheless, the conjecture is that Chow groups are finitely generated - this is called ``refined Bass conjecture'' in these talk notes of Thomas Geisser. In those same talk notes, you can also find Parshin's conjecture, which would imply that $CH^2(X)$ and $CH^3(X)$ are not just finitely generated but also torsion (if the base field is finite). Provided you believe the homotopy classification of projective modules above, the refined Bass conjecture and Parshin conjecture imply the weak Bass conjecture for $A$ finitely generated of Krull dimension $\leq 3$ over $\mathbb{F}_q$.

Some further remarks:

1. The counterexample proposed in Daniel Litt's comment would not be a counterexample if the refined Bass conjecture and Parshin conjecture are true. For a smooth affine surface over $\mathbb{F}_q$ with trivial Picard group, all the information would be in $CH^2(X)$ (maybe an oriented Chow group would be needed). So, although there are many $0$-dimensional local complete intersections in the affine surface, the isomorphism type of the vector bundle only depends on the class in $CH^2(X)$ - and there would only be finitely many such classes.
2. Generally, the classification of projective modules will be impossible to do, much like it is impossible to classify vector bundles on compact manifolds in general. Nevertheless, the expectation is that there is a range (called metastable range by Suslin) in which the projective module classification can be done in terms of Chow groups, K-theory, hermitian K-theory and operations connecting them. In this range, finiteness of projective modules (over function rings of smooth affine schemes over $\mathbb{F}_q$) would follow from finite generation of Chow groups (refined Bass conjecture) and Parshin's conjecture.
3. The extension of the above argument to finitely generated $\mathbb{Z}$-algebras would be a lot more subtle. It would need the full Bass-Quillen conjecture on projective modules over polynomial rings. Moreover, the $\mathbb{A}^1$-homotopy classification results for projective modules have not been done (though that obstacle seems a lot smaller than the general Bass-Quillen conjecture in the non-geometric case). It is also not clear to me if the resulting motivic cohomology groups in arithmetic situations would be expected to be finite, or just finitely generated. A search for counterexamples in such settings might be more promising.

First of all, I am a bit confused about the finitely many isomorphism classes statement. The finite generation for the Picard group is known (which implies finiteness of $K_0(A)$ in the case of curves over finite fields), but this does not imply that there are finitely many isomorphism classes.

Second, by a result of Serre, if $X$ is an affine variety of dimension  $d$, then all vector bundles of rank $>d$ split off a free direct summand. The question for arbitrary rank $n$ then reduces to discuss ranks $\leq d$. In particular, the question has a positive answer for curves.

Now, I would like to discuss the $\mathbb{A}^1$-homotopy approach to the classification of projective modules, and outline how one might use that to understand the relation between the Bass conjecture and the finiteness of projective modules.

The first comment concerns the UFD requirement. The finite generation for the Picard group is known. This implies a positive answer to the Bass conjecture for $K_0$ in case of curves over finite fields. Using that the Picard group is finitely generated, I think you can replace the UFD requirement by classifying projective modules up to isomorphism and tensoring by line bundles.

Second, the weak Bass conjecture implies the Bass conjecture: by a result of Serre (Theorem 1 in J.-P. Serre: Modules projectfs et espaces fibrés à fibre vectorielle. Séminaire Dubreil, Dubreil-Jacotin et Pisot, 1957/1958, Fasc 2, Exposé 23, p. 18, Sécretariat mathématique, Paris, 1958), if $X$ is an affine scheme of Krull dimension  $d$, then all vector bundles of rank $>d$ split off a free direct summand. But this means that there is a global upper bound for the number of projective modules of given rank $n$, namely the sum of the numbers of projective modules of rank $\leq d$. The weak Bass conjecture would say that this is a finite number. But then, the group completion of the monoid of isomorphism classes of projective modules will be finitely generated, so we have the Bass conjecture.

Now, as a third point, I would like to discuss the $\mathbb{A}^1$-homotopy approach to the classification of projective modules, and outline how one might use that to understand the relation between the Bass conjecture and the finiteness of projective modules. In particular, the arguments below should convince you that the weak Bass conjecture holds for smooth affine varieties of dimension at most $3$, provided a refined version of Bass' conjecture and the Parshin conjecture hold.