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Technically, the RAP circuit on the H3 does NOT control the radio, it supplies power to components such as the sunroof, wipers, windows, and the power is supplied through separate fuses that obtain power from the RAP/ACCY relay. This relay is controlled via a ground wire by the BCM; which as you know, shuts down power about ten minutes after the ignition has been turned off, or a door has been opened or closed after the ignition has been turned off.
The Radio, as well as a few other components, is not part of the relay controlled RAP circuit. The radio is fed constant battery voltage that serves two purposes. One, keeps any memory settings that are not contained on a non-volatile memory chip, and is used to power the radio when the turn on signal is received from the BCM.
When you start the vehicle, the BCM sends a turn on signal to the radio via the class 2 serial data circuit, and the radio wakes up and goes to the last position you had set on the radio (on/off, CD/radio, etc.). After you turn off the ignition, the BCM will shut down this circuit at the same time it removes the ground to the RAP relay.

That's how the two circuits operate. Is there a way to override the RAP circuit? I would say yes, but it would depend on obtaining a signal from the module used to remote start the vehicle, and I am not familiar with that item. As for the radio circuit, that would require interfering with the serial data circuit, which, unless you know exactly what you are doing, I would not touch. Consider this to be the internal local area network where all the computers talk to each other. (This is the circuit where the BCM sends the security ok to the PCM, this is the circuit that verifies VIN numbers of components, etc.)