Some thoughts I had while reading this, probably not very original: the fact that chess moves are not random and usually adhere to opening theory to some degree means you could get some value out of using something like a trie of some small finite depth, trained on a database of games, to compress common opening lines much better, at the cost of making things like 1. a4 a5 2. h4 h5 that aren't in the trie much costlier to represent. A practical way to actually do this would likely look like applying arithmetic coding to all of the possible length-N prefixes of the game.

The mid- or lategame are also far from random and could probably be compressed in small "chunks" via a coder that would (effectively) learn to predict patterns of e.g. capture being followed by a recapture and cause such chunks to require fewer bits to represent.

I'm not very knowledgeable about compression algorithms, though; I'm sure others will be able to provide corrections or references.