This is not a full answer to the question but it was too long for a comment.
Depending on your conventions, "weak $n$-categories" might mean "$(n,n)$-categories" which are a special case of $(\infty,n)$-categories. If this is indeed the case, some things can be said. First, the question also makes sense for $\mathbb{E}_n$-monoids which (upto completion) may be thought of as $(\infty,n)$-categories whose space of objects is connected. Let me focus on that case.
In my Msc. thesis1 I study the interplay between coherence and arity for $\cal O$-monoids where $\cal O$ is an $\infty$-operad and arity is, roughly speaking, the $k$ in $x_1 \otimes \cdots \otimes x_k$. In particular I show that the coherence data for $k$-truncated, $\mathbb{E}_n$-monoids is concentrated in arities $\le k+3$. Combining this with the fact that the $\infty$-operad $\mathbb{E}_n$ has finitely many cells in every arity, it follows that specifying an $(n,n)$-category with a connected space of objects involves only finitely many coherence diagrams.
Unfortunately, I don't have anything definitive to say about the general case. However, the tools I use seem very robust and I have no reason to believe that similar techniques might not be applicable beyond the connected case. I haven't attempted this though, as it is quite far from the original application I had in mind.
1 A preprint will soon appear on the Arxiv.