Hi John,
During the span of time multiple models of BMC radios were availability in a number of cars. I would suggest you be very careful that the radio you have can be polarity switched. About a year after the delivery of my new BJ8 in 1964, I installed a new AM BMC radio that remained installed until converting to negative ground about 20 years after. At that time, I attempted to convert the original BMC radio but found, verified by an expert, that the polarity could not be easily changed. The reason given was as has been mentioned, that the case the ground for the radio as is the antenna. Modification would require the internal boards be changed substantially to accept the change. As you indicated, other models of BMC radios had the internal circuitry isolated from the case and, as such, could be altered to operate in a negative ground environment.
I am not sure what BMC model radio you have but I would be very careful thinking it will be easy. Although I still have my original BMC radio, when changing car polarity, I decided it would be far safer if I acquire a negative ground radio that would accept the clear plastic BMC display and install it instead of risking and trying to modify the original. The original has remained on the shelf for over 30 years and I will probably never reconvert my car back to positive ground. The replacement negative ground radio, with the BMC display, is still installed in my original console and is also sitting on the shelf waiting for the time when I ever think of going back.
Bob may be correct and your model may be easily convertible...or maybe not.
Good luck,
Ray (64BJ8P1)