If the bat only cost $20, leave it outside the box, next to the box- ventilation problem solved. If the bat is a SLA 12V 7A bat, it will survive the heat with no problem. I have a couple SLAs that I use for radio control and power loss lighting/radio that stays in my garage. I live in the desert where the outside temp gets up to about 120 deg F or 48.9 deg C. The garage reaches temp of 130+ F. These bats are three years old and still work great.
Bottom line, put the circuit in the box, the bat next to or on top of the box and the whole thing in the shed. If you run into a problem, bury the bat. and worry about water, rain, solvent spillage, and rodents chewing up the underground stuff.
I would use an old paint can to put an alarm system into so if someone broke into the shed first, they wouldn't "recognize" the box right away.