The car uses a BMW engine and gearbox, plus power steering and A/C.
The various control modules are modified by MMC and they are wired into a CANBUS loom from a 1990s BMW 3 series, I understand.
The V8 was never installed into the 3 series, so there are MMC adaptations to the CANBUS Loom so that the engine can talk to the Morgan instruments and work with the Morgan ABS and Morgan airbags.
It is, in all honesty, a bit of a dog's dinner.

I don't think MMC has ever understood why the car drains power so, I was told that the Manual cars don't, so I assume that it is something to do with the Auto Box/Paddle shift control modules.
It isn't right and if it was a mainstream manufacture I'd expect answers.
But it is Morgan.and essentially a car that was developed in the early 2000s: with a charger I can live with it.

Could you run a separate (in-line fused) live feed to the radio, to keep it alive even if you turn off the master switch ?

Jon, that's how both of my Plus 8's were wired which makes sense to me.
The 4/4 Sport didn't but no radio or Master switch until I fitted the later.

Do the new CX Morgans have any similar issues?

A good question... !!
Next question.

I think as the years roll by, lots of CX owners will have electrical gremlins

My manual does the same, so on a conditioner all the time. Dogs dinner it is.

It is nice to know I'm not alone.....

It is a feature sir

Everyone knows MMC use cheaper alternatives. They appear to be putting cheap electrons into the cars.