The other constraint is that the pawn position must be proper. Besides no pawns on first rank, there's no pawns on eighth rank. There are some corollaries about when a file can have pawns in unusual order - if nothing's captured, then on every file, the white pawns must be in front of the black pawn. This makes me think that the most numerous class of boards are those with one or two captures and one or two promotions. These only matter in the initial position.
Thes provide a few other extremely rare constraints.
No triple checks. The first one I could think of was a king behind the enemy' s contiguous row of pawns. But it seems like lots of them couldn't just be proven impossible by rules, and could only be weeded out by exhaustively checking the previous line of play to see if the board was reachable. In other words, each of us should always send any new comments or messages to the other two, so all three of us will always be up to date. I have some messages from each of you that I assume the other has not yet seen.