Sounds like a theft system problem for sure.
Very possibly a faulty GEM module(Generic Electronics Module) which is the body control module.
Have the system data scanned for trouble codes using an advanced system scanner before throwing parts at this.
I suggest taking this in to the dealer or qualified repair facility for diagnosis first and then decide if you can repair yourself or just have them do it.
Some GEM modules need to be programmed to the vehicle when installed.I dont know for sure on this year or not.
Programming needs to be done by the dealer or a shop with the capabilities for this.