Related to @BlackHatCoiner idea, the avatars could be chosen on a first-come, first-served basis, or perhaps by a lottery in which all members participate?
My meritocratic character suggests that they should be spread based on earned merit. In that case, fillippone would get the white king (or deny it, since he's wearing the Foxpup avatar). The other king would be icopress. Next, the white and black queen, which would go to cygan and dkbit98 respectively, followed the rooks, GazetaBitcoin and 1miau, etc.
A provably fair lottery using block hashes could also determine which piece each of the 31 of you gets to have. Whatever icopress prefers. However, if we go with the lottery, then let me know, because it has to be set up in a particular way, so that certain bias is avoided, and each participant gets an equal probability to win.