I don't have a complete answer, but here are some thoughts.

The Rips Construction takes an arbitrary finitely presented group Q and produces a 2-dimensional hyperbolic group $\Gamma$ and a short exact sequence

$1\to K\to \Gamma\stackrel{q}{\to} Q\to 1$

such that the kernel $K$ is generated by 2 elements. It turns out, by a result of Bieri, that $K$ is finitely presentable if and only if $Q$ is finite.

One can improve the finiteness properties of $K$ using a fibre product construction. Let

$P=\{(\gamma,\delta)\in\Gamma\times\Gamma\mid q(\gamma)=q(\delta)\}$.

By the '1-2-3 Theorem', if $Q$ is of type $F_3$ then $P$ is finitely presentable.

I would guess that $P$ has good higher finiteness properties if and only if $Q$ is finite. Perhaps one can use the fact that $P\cong K \rtimes\Gamma$.

Even if this is true then it still doesn't quite solve your problem, as we don't have a presentation for $P$. To do this, one needs to be given a set of generators for $\pi_2$ of the presentation complex of $Q$, which enable one to apply an effective version of the 1-2-3 Theorem. (In the absence of this data, presentations for $P$ are not computable. Indeed, $H_1(P)$ is not computable.)

Question: Does there exist a list of presentations for groups $Q_n$ such that:

1. each group $Q_n$ is of type $F_3$;

2. the set $\{n\in\mathbf{N}\mid Q_n\cong 1\}$ is recursively enumerable but not recursive;

3. but generators for $\pi_2(Q_n)$ (as a $Q_n$-module) are computable?

If so, and if I'm right that the higher finiteness properties of $P$ are determined by $Q$, then higher finiteness properties are indeed undecidable. Simply apply the Rips Construction and the effective version of the 1-2-3 Theorem to the list $Q_n$.