Since the mice run 2xAA in series, you might get away with running one of those and jumping the other cell space.

If 3.7-4.2V is too much to handle, jump with a 1n4148 diode or two in series.

My Logitech G604 takes only one AA. I'm not sure the advantages over NiMH (lighter weight, higher capacity) are worth soldering in diodes especially when the efficiency loss would negate the capacity advantage.