It sounds to me you seek a solution to the following problem:

> Given a suitable subcategory of $\textbf{Set}$, find a Grothendieck topology on $\textbf{Set}$ so that sheaves on $\textbf{Set}$ are equivalent to presheaves on the subcategory.

Fortunately, this can be done in great generality.

**Theorem.**
Let $\mathcal{D}$ be a locally small category and let $\mathcal{C}$ be an essentially small full subcategory of $\mathcal{D}$.
Then there is a unique Grothendieck topology on $\mathcal{D}$ with the following property:

* Given a presheaf $F : \mathcal{C}^\textrm{op} \to \textbf{Set}$, there is a unique (up to isomorphism) sheaf $\tilde{F} : \mathcal{D}^\textrm{op} \to \textbf{Set}$ whose restriction to $\mathcal{C}$ is $F$.

**Example.**
Let $\mathcal{D} = \textbf{Set}$ and let $\mathcal{C}$ be the full subcategory of sets of cardinality $\le k$, where $k \ge 1$.
Then the hypotheses of the theorem are satisfied.
The Grothendieck topology so obtained can be explicitly described as follows: a family of maps to $D$ is covering if and only if, for every map $f : C \to D$ where $C$ has cardinality $\le k$, $f$ factors through some member of the family.
Put it even more simply, a family of maps to $D$ is covering if and only if every subset of $D$ of cardinality $\le k$ is contained in the image of some member of the family – which agrees with the answer of Tom Goodwillie here.

*Proof of theorem.*
Consider the restriction functor $j^* : [\mathcal{D}^\textrm{op}, \textbf{Set}] \to [\mathcal{C}^\textrm{op}, \textbf{Set}]$.
Clearly, it preserves finite limits.
Let $j_* : [\mathcal{C}^\textrm{op}, \textbf{Set}] \to [\mathcal{D}^\textrm{op}, \textbf{Set}]$ be right Kan extension along the inclusion, explicitly:
$$(j_* F) (D) = \textrm{Hom} (j^* h_D, F)$$
Then $j_* : [\mathcal{C}^\textrm{op}, \textbf{Set}] \to [\mathcal{D}^\textrm{op}, \textbf{Set}]$ is the right adjoint of $j^*$.
Given $F : \mathcal{C}^\textrm{op} \to \textbf{Set}$ and $C'$ in $\mathcal{C}$, we have
$$(j_* F) (C') = \textrm{Hom} (j^* h_{C'}, F) = \textrm{Hom} (h_{C'}, F) \cong F (C')$$
by the Yoneda lemma.
Hence, $j^* j_* F \cong F$, so – modulo some checking that this actually is the counit isomorphism – $j_* : [\mathcal{C}^\textrm{op}, \textbf{Set}] \to [\mathcal{D}^\textrm{op}, \textbf{Set}]$ is fully faithful.
Thus – modulo size issues – the hypothesis means $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ is equivalent to a subtopos of $[\mathcal{D}^\textrm{op}, \textbf{Set}]$, with $j_*$ being identified with the inclusion.
But subtoposes of a presheaf topos exactly correspond to Grothendieck topologies, so we are done (modulo size issues).

Explicitly – and this is how we check that there are no size issues – given a sieve $U$ on an object $D$ in $\mathcal{D}$, consider it as a subpresheaf of the representable presheaf $h_D$ and define it to be $J$-covering if $j^*$ sends the inclusion $U \hookrightarrow h_D$ to an isomorphism $j^* U \to j^* h_D$.
Very explicitly, that means $U$ is a $J$-covering sieve on $D$ if and only if, for every object $C$ in $\mathcal{C}$ and every morphism $f : C \to D$ in $\mathcal{D}$, $f \in U (C)$.
It is not hard to verify that $J$ so defined is a Grothendieck topology on $\mathcal{D}$.
Given a presheaf $F : \mathcal{C}^\textrm{op} \to \textbf{Set}$ and a $J$-covering sieve $U$ on $D$, we have
$$\textrm{Hom} (U, j_* F) \cong \textrm{Hom} (j^* U, F) \cong \textrm{Hom} (j^* h_D, F) \cong \textrm{Hom} (h_D, j_* F)$$
so $j_* F$ is a $J$-sheaf on $\mathcal{D}$, and $j^* j_* F \cong F$ as we previously noted.
Thus, $J$ is the required Grothendieck topology.　◼