Wouldn't it be possible to use some kind of passive heat transfer to at least match the ambient temperature? I don't know what one would look like, but I'm visualizing something like a computer heatsink built into the side.

It could also be vented somehow, with the vents shielded from the elements.

There are also water cooling kits for computers... Possibly something like that could be used on the cold water line / hot water from the tub.

Hmmm.... There are a lot of possibilities, its just hard to figure out what is feasible....

How waterproof is the remote BTW?

-Biscuits