How do I replace this bike light cache battery?
Since I can't find a suitable replacement cell, I think the best answer is to find an easily obtainable lithium phosphate cell pair and wire them up, build an external piggyback for the light, and wire it in.
Something like a AA size (14500) LiFeO4 cell seems to be pretty common as they're used in solar lights, and they're inexpensive. In the AA size, it would be simple to get a 2xAA battery container, wire it up, waterproof it, and call it done.
At a cell price of 4 for $16, and a holder price of 4 for $12, that would be a simple and cheap solution in perpetuity, and wouldn't require any re-jiggering of the electronics.