Batteries should not be needed on the H200 or H310, they are not caching controllers. (And even if they were, that is not something that will work well with ZFS.)
Anyway, yes, it is the heat that concerned me. Even with good airflow, less heat created is better.
Both work in a standard PCIe slot. The main difference is the location of the SAS connectors. On an H200, they are near the bracket, easy to get to in a standard case. The H310 has them inline at the far end of the board, which might be better in a true server case. Another thing to note is that Dell has some of these that are a nonstandard card with a different form factor, made for matching slots on some Dell servers.
Well hopefully they designed the H200 to work well within the R710. I'm a noob at this so I have no clue how all this stuff is designed. If it burns up then I'll follow your advice and go get a H310.