= Solution
A <coverage on a category> specifies covering families stable under refinement along arrows: if $(f_i:C_i\to C)$ covers and $h:B\to C$ is any arrow, there is a covering family $(g_j:B_j\to B)$ such that every $hg_j$ factors through some $f_i$. Equivalently one works with the associated saturated covering <sieves on a category>, satisfying maximality, pullback stability and local transitivity. These determine the same <sheaves on a site>. We use this <sieve on a category> formulation of $J$.
An object is a <J-irreducible object of a site> if its only covering <sieve on a category> is the maximal <sieve on a category>. The <coverage on a category> is a <rigid coverage> if every object is covered by the <sieve on a category> generated by arrows whose domains are J-irreducible.
Assume $\mathcal C$ has finitely many arrows and every <idempotent morphism> splits. There are finitely many <sieves on a category> on each object $C$, so the intersection of all covering <sieves on a category> is a covering <sieve on a category> $R_C$: finite intersections of covers are covers by the topology axioms. It is the least cover. Let $I$ be the collection of all arrows $f:B\to C$ lying in $R_C$. This is an <idempotent ideal in a finite category>.
Here are the details. It is closed under precomposition because $R_C$ is a <sieve on a category>. For $h:C\to D$, the <sieve on a category> $h^*R_D$ covers $C$; minimality gives $R_C\subseteq h^*R_D$, so $hf\in R_D$. Thus it is also closed under postcomposition. Local transitivity says that the <sieve on a category>
$$
S_C=\bigcup_{f\in R_C}fR_{\operatorname{dom}f}
$$
covers $C$. It is contained in $R_C$, while minimality gives the opposite inclusion. Hence $S_C=R_C$, or $I^2=I$.
We prove that such an ideal is generated by the identities it contains. Take $f\in I$. Since $I^2=I$, repeatedly factor its remaining right factor to obtain
$$
f=a_1b_1=a_1a_2b_2=\cdots=a_1\cdots a_nb_n,
$$
with every factor in $I$. Write $p_n=a_1\cdots a_n$. Finiteness gives $p_i=p_j$ for some $i<j$. Their domains coincide, and $t=a_{i+1}\cdots a_j$ is an endomorphism in $I$ satisfying $p_it=p_i$. Some positive power $e=t^m$ is idempotent: the powers are eventually periodic, so choose $m$ beyond the eventual threshold and divisible by its period. Then $p_ie=p_i$. Split $e=ir$ with $i:B\to\operatorname{dom}(p_i)$, $r:\operatorname{dom}(p_i)\to B$ and $ri=1_B$. Since $I$ is a two-sided ideal,
$$
1_B=rei\in I,\qquad f=p_i e b_i=(p_i i)(r b_i).
$$
Thus $f$ factors through an object $B$ whose identity lies in $I$. The latter means $1_B\in R_B$, hence $R_B=yB$; every covering <sieve on a category> on $B$ must therefore be maximal. So $B$ is J-irreducible.
Every arrow of $R_C$ consequently factors through a J-irreducible object. The <sieve on a category> generated by arrows from all such objects contains $R_C$ and is a cover. This proves \b[every coverage on the finite idempotent-complete category is rigid]. If $R_C$ is empty, the same conclusion holds because the empty <sieve on a category> already covers $C$.
Let $\mathcal D$ be the full subcategory of J-irreducible objects. For a covering <sieve on a category> $S$ on $C$ and any $f:D\to C$ with $D\in\mathcal D$, the pulled-back <sieve on a category> $f^*S$ covers $D$ and is maximal, so $f\in S$. Therefore $S|_{\mathcal D}=yC|_{\mathcal D}$. For a <categorical presheaf> $F$ on $\mathcal D$, define its extension by
$$
H_F(C)=\operatorname{Nat}(yC|_{\mathcal D},F).
$$
This is the right Kan extension along the inclusion. Its adjunction gives
$$
\operatorname{Nat}(S,H_F)\cong\operatorname{Nat}(S|_{\mathcal D},F)=H_F(C),
$$
which is precisely unique amalgamation on $S$. Hence $H_F$ is a sheaf. The <Yoneda lemma> gives $H_F(D)\cong F(D)$ for $D\in\mathcal D$.
Conversely, if $G$ is a sheaf, its sections on all arrows from $\mathcal D$ into $C$ determine a section on their covering <sieve on a category>. A compatible family extends to an arrow factoring through such a $D$ by restriction from its given $D$-section. If an arrow has two factorizations, restrict to the covering <sieve on a category> generated by arrows from $\mathcal D$ into its domain; compatibility makes the two restrictions equal, and the sheaf's uniqueness of amalgamation makes the sections equal. The resulting matching family glues uniquely over $C$. Thus $G(C)\cong\operatorname{Nat}(yC|_{\mathcal D},G|_{\mathcal D})$. Restriction and $F\mapsto H_F$ are inverse up to <natural isomorphism>, proving
$$
\boxed{\operatorname{Sh}(\mathcal C,J)\simeq[\mathcal D^{\mathrm{op}},\mathbf{Set}].}
$$
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