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Why not create a few standard formations, say swap the Bs & Ns, or put both
Ns on one side and both Bs on the other, that you choose randomly for each
player. Place the Ks and Qs in the middle 2 squares, again randomly for
each, short and long castling depending on each king's initial position.
So you've got RNB, RBN, RNN, and RBB, and with the K and Q random,
there's 8 possibilities/side, or 64 different opening setups. All are
close, I think, and might be balanced enough to work. If not, pick the 32
most balanced, and just use those. The generalized opening strategy for all
possibilities should be fairly obvious, but I suspect the individual
details would be sufficiently different that the memorization of the first
20 moves of a few hundred openings would be much less an advantage.
However, all of a player's actual chess ability is totally untouched by
this scheme, and the general opening principles are already very well
established. If you wish to get out into the deep end here, you could play
a game where one side had all 4 knights and the other, all 4 bishops, or
add both possibilities into the mix and strain people a little bit. ;-)