I wouldn't include shuttles or runabouts and instead use the Miranda class as Pawns.
Kings should be some for of station - that would explain their strategic purpose in the game, as well as their (relative) lack of motion. Either something like the Earth Spacedock or DS9 (yes, it is a Cardassian design, but it looks cool). For Queen, I would go with the Galaxy. I agree with those other assignments you have for the Federation side, though perhaps the pawns can be of the Oberth class instead of the Miranda. Also, for some reason I feel like the Excelsior would be more of a Bishop and the Nebula wold be more of a Rook.