LiFePO4 battery in sealed container
Nothing wrong with LFP batteries. Their beauty is that they don't blow up or burst into flames when things go wrong. That said, what could go wrong...?

Why are you worried about temperature? 
If they are getting hot, you're doing it wrong.

Perhaps a wide-ish PVC pipe, the same inner diameter as the battery length (or batteries), going straight down. The one end is given a properly sealed end-cap with PVC weld. The other end can be on the surface, with a big, plumbing screw-on cap. 

Holes are drilled in the cap, wires pulled through, and you silicone around that wires to protect from splashes.

But going back to the battery, if weight isn't a terrible issue, a 12V 10AH SLA battery would be perfectly fine. It would likely have similar weight to an LFP.

Stick on a 20W panel and a solar controller, and you have power sorted. Even if it couldn't make the thing fully "unplugged", it would extend the running time well, wouldn't be difficult to implement, and would be roughly the same price as similar sized LFP batteries.