Is there a polynomial sized arithmetic formula for iterated matrix multiplication?
After reading over everything more carefully, I think I was misunderstanding the argument a bit, and as Yuval points out, IMM can be computed transparently in poly size without having a poly size formula. The idea seems pretty straightforward too:
Everything stated previously applies to all rings, so if the elements of the matrix were $Z_2$, then we could consider the ring of $n \times n$ matrices over $Z_2$. There's clearly a $\log n$ depth formula for the product of $n$ ring elements, so we can apply theorem 4.
The only additional considerations are

Theorem 4 says the program will need only 3 working registers, but since each register holds an entire matrix, these registers will take up $O(n^2)$ space. This turns out to be completely fine for us, since we have polynomial catalytic space to work with.

In order to actually translate the program back onto our (catalytic) turing machine, we need to rewrite matrix instructions to instructions on the underlying ring ($Z_2$ for example). Luckily, this is trivial since each instruction in the form $r_i \leftarrow r_i \pm x \ast y$ becomes $n^3$ primitive instructions: $r_i^{uv} \leftarrow r_i^{uv} \pm x^{uw}y^{wv}$ for $u,v,w = 1 \ldots n$ and no extra space of any sort is used.


This seems to be roughly in agreement with the $O(n^9)$ time complexity given by the second reference from the OP -- the program size of $4^d$ from Theorem 4 combined with the $\log_2 n$ depth of the straightforward matrix multiplication formula makes an $O(n^2)$ sized program, each matrix instruction which is further broken down into $O(n^3)$ &quot;primitive&quot; instructions, and each instruction might take up to $O(n^3)$ time to execute on a turing machine due to the size of the inputs.