Something to note - the battery is likely AGM instead of Lead Acid and doesn't like being charged at higher voltages so hopefully you have a charger that can charge AGM batteries.
One other somewhat common issue is a bad ground strap in the engine bay. There should be some way to bypass it to check if that is it. Here's another thread related to that -
https://www.e90post.com/forums/showthread.php?t=1441893
I would think if its transfer case you would have some signs leading up to it like feedback in the steering wheel while turning in parking lots or something. Also if its the transfer case I would think the car would start fine and maybe even run in RWD only.