[Arran]

Killing radio reception is rather easy, you only need to short the antenna input to ground, this set has no tuned RF amplifier stage, so there isn't a duplicate set of coils to deal with. Another other method would be to switch off the oscillator. In this set it uses a separate detector/AVC tube so you can cut the RF signal there Then again you can usually find a dead spot on the dial, on one of the bands, where nothing is going to interfere with the audio from the Bluetooth.
As Mr Fix points out, and I had forgotten that this era of RCA/CGE sets had this, there is a screw terminal board on the back with a link that cuts the connection between the 6H6 detector/AVC tube, and the 6F5, presumably the switch to engage, disengage the radio was built into the record player attachment, which you can duplicate. Because that phonograph terminal is already there I would not go poking around in the band switch, that is asking for trouble as the switches are complex, and almost unobtainable if you break something, and many of the terminals are not easily accessible, it's just not worth the risk to save a few Watts of electricity. The Australian fellow, Don, of Don't Old Radio shed has added Bluetooth to a number of sets he's worked on, and I think he had it set up to electronically switch in the Bluetooth, I myself would prefer a Part 15 transmitter to broadcast what I want without altering the radio, but there is still enough content on AM around here that I can't be bothered, at least at night.
Regards
Arran