I had something similar, but not quite the same, on my 2002 FFRR.
My screen worked properly the whole time, but often the sound would switch off after a few minutes. I never really resolved whether it was a fixed amount of time, but I began to feel that it might be related to going over a bump in the road. Not always, but sometimes that seemed to be the case.
In mine, there was no sign of water ingress that affected all the electronics in the boot LH side, but I did discover dampness in the spare wheel well, eventually traced to a badly fitted tow bar. I guess this would have increased the moisture levels overall, but not actively got the electronics wet apart from the air suspension compressor, which I had to replace.
Eventually I solved the radio issue - and kept the car another 3 or 4 years, so really confident it was solved - by taking apart the BM54 module and carefully resoldering as many of the joints on the circuit board as I could. It appears that there was at least one dry joint there - and in my experience, if there is one there will usually be many.
If you're not so confident about your PCB soldering, I imagine there are companies around who would do this for you, or perhaps offer and exchange BM54. These units are fairly widely used on BMW vehicles as well, so not uncommon. At a push, I suppose, you could try a second hand BM54 from eBay, although that might have just as many problems as your own and you simply can't tell.
Not sure if this is helpful at all..... Current: 2010 Range Rover L322
Vogue SE, TDV8 3.6, Bournville
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Now gone: 2002 Range Rover L322
Vogue, TD6, Giverney Green
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