I haven't got the Fuji X-M1 but interestingly My SONY A77v uses less power to the back screen than the Electronic Viewfinder, states it so in the Manual. It recomends using the screen to save battery.
I get 4-500 frames easily regardless and I have two batteries so never a problem.
As Ken says, the battery normally lasts longer than me.
I wouldn't worry about the heat problem either, internals are quite different from any DSLR arrangement. Again my SONY doesn't seem to suffer this issue at all even after along night.