an LLM would complain that their internal model does not refelct their current input/output.

Since LLM's knows people knock off/test/run afoul/mistakes can be made, it would then raise that as a possibility and likely inquire.

This isn't prompt engineering, it's grammar-constrained decoding. It literally cannot respond with anything but tokens that fulfill the grammar.

A grammar is really just a special case of the more general issue of how to pick a single token given the probabilities that the model spits out for every possible one. In that sense, filters like temperature / top_p / top_k are already hacks that "do the work" (since always taking the most likely predicted token does not give good results in practice), and grammars are just a more complicated way to make such decisions.

I'd be more interested in what the distribution of grammar-restricted predictions looks like compared to moves Stockfish says are good.