This looks great, looking forward to reading it in more detail!
Just FYI, the partial dot product notation Argyris and I used in our proof was generally disliked by referees, so we've updated our paper to not use it (will be updated on arXiv at some point).
Instead of partial dot products, you can write this as an operator acting on a vector
$$
\vec v_2'\odot(\vec v_1\otimes\vec v_2) =(I_1\otimes \vec v_2'^T) \cdot (\vec v_1\otimes\vec v_2)
$$
which people seem to like better.
"All cluster states are local-Clifford equivalent to a stabilizer state" Did you mean the reverse: all stabilizer states are local-Clifford equivalent to a cluster state? Cluster states are already stabilizer states; the more interesting fact is that (as your GHZ example illustrates) an arbitrary stabilizer state can be locally Clifford deformed to a cluster state.
Just FYI, if you've got a BSM that works >66% of the time, you can do fault-tolerant quantum computation with *unencoded* 6-ring resource states, which are a lot more feasible to generate, see https://arxiv.org/abs/2301.00019.
There's also some subsequent discussion of this construction (and a few others) in https://arxiv.org/abs/2303.16122