Sounds like stuff on the retained accessory power part of the BCM, so the BCM is very likely. Haven't delt with anything quite like that first hand, hmmmm
So what I would do is check the circuitry from the BCM to the fuse panel, then disconnect the aftermarket radio and see if it still acts up, if it does it's most likely a BCM, but without checking it firsthand I don't want to tell you to run out and buy a BCM. At least those BCMs are not a big deal to install, once they are installed they learn the theft stuff on the first key-up (if you get a new one) I'm not sure about getting a used BCM, but you'd at least need to get the correct key resistance for it to start the car.