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...