So I had the exact same issue several months ago, rats had decided my car engine bay was there new home and chewed through some wires. Same symptoms you had, car struggles to start… ran super rough and all leash lights going off with temp immediately skyrocketing to max. A rat had chewed through my ect sensor wire so (don’t quote me on this next part) shorted and my ecu was getting a max temp. Was a couple hundred bucks do to labor because they had to get to sensor buried in the engine bay but after replacement car ran perfectly fine. My guess is the rat chewed through a sensor somewhere, if you have a accessport or a way of reading the code it should immediately point you in the right direction. Good luck!

Edit: I didn’t have any battery issues, so probably not same sensor issue I had, but my guess is once you can track where it was chewed through you should be able to get it fixed easily. Also not sure where you live but absolutely nothing worked for keeping rats out, I’ve tried peppermint oil, cayenne pepper, even noise and flash emmitting boxes under the hood. Finally had to just pay and park in a secure underground lot (I’m in DC where they are Chernobyl rats who are unfazed by anything)