@mangledorf Sorry, I think the right term is rank-revealing QR decomposition. This presentation explains the method and its application to this problem.

@Joël Is "almost all finite groups are $2$-groups" actually known? I was under the impression it was still open.

I have tried using Sage and Magma, but they are not able to determine membership in a f.g. matrix group over $\mathbb Q[\sqrt2]$.

@Matheus Whoops, appears I made a miscalculation. The surfaces $V_{2^m}$ are also perfect, so it even answers my more general question.

@Matheus Yes, this is precisely what I was looking for. If you post it as an answer I'll accept it. Unfortunately it doesn't generalize to other $V_{2^m}$ as these are not perfect surfaces, but it answers this question completely. A possibly interesting side note: it seems that authors in irrational billiards use "combinatorial type" exclusively, while in the rational literature "cutting sequence" is more common.