The words 'a fairly standard set' set me off wondering how easy it could be played with distinguishable FIDE sets. With 1 aside of 6 piece types, 2 aside of 11, and 4 and 16 aside of the remaining 2, 3 sets cannot do it, but 4 can. I began by assuming that 2 sets' Pawns would be ignored, but then realised that it would make sense to use half of one set's Pawns as Alfils. When I laid everything out I realised that the last set had only two piece types in use. This made me wonder whether I could substitute a more exotic set for the third and fourth. A Xiang Qi set would again not quite do it, as using only four Points aside would mean only 15 out of 16 pieces. A Shogi set, however, did the trick and I finally hit on the following representation:
FIDE set 1 - all pieces as themselves.
FIDE set 2 - King as Bat; Queen as Nightrider; Rooks as Tigers; Bishops as Camels; Knights as Giraffes; Pawns as Pawns.
Shogi set - Rook as Gryphon; Bishop as Rhino; paired pieces as Wazirs, Ferzes, Men (aka Princes), and Walkers (aka Stewards); Points promoted as Dabbabas and unpromoted as Alfils.