Isn't this vehicle fitted with dual batteries, one for starting and one for vehicle operations. With that said, did the correct battery get replaced? There may be some bulletins relating to the BCM (Battery Control Module), that may be a good avenue to investigate as well.

If it were me the first thing I would check is the main connecting circuits, I would confirm the integrity of all the power and ground circuits by conducting a number voltage drop tests. Lots of current needed to operate many different things, one loose connection and it is anybodies guess what symptoms will evolve.