Thanks for the info - sounds like a very professional job!
I think I've sorted the problem with the radio:
I was convinced it was the car doing something odd - but this turned out not to be the case.
I checked the voltage drop using the rear power socket as feed for the radio:
Engine off, radio off, 12.03v. radio on receive, 11.99v. Tx on low power (5 watts) 11.72v, Mid1 11.63v, Mid2 11.49V, high (50 watts) 11.04v.
Engine on, radio off, 14.6v. radio on receive, 14.6v. Tx on low power 14.34v, Mid1 14.28v, Mid2 14.14V, high 13.8v.
So, only dropping about a volt with a 50 watt transmission - I could have lived with that, but I decided to go to a better supply point to try and sort the intermittent radio problem.
I took the feed from the big red cable in the passsenger footwell, and created a new earth connection down in that corner for convenience, good quality heavy twin feedline, fused in both 12v and earth. Got that all sorted, powered up the set and it was fine - for about 5 minutes. It again powered itself off, and refused to switch back on. (I was not a happy bunny at this point!)
Nothing I tried brought the radio back to life - I was still convinced the problem was something to do with the car's electronics, till I tried supplying the radio from a separate 12v gel-cell - and still had the problem. (As you held the power button down, there was a sort of "ticking" noise from the speaker).
A bit of digging on the internet revealed that others had seen the same issue with the FT-7900 - the investigations got rather involved, but stem from the internal circuitry not latching a main 13.8v supply line in the radio. There was a suggestion that it could have something to do with the cable connecting the radio's head unit back to the main box. I dug out the short lead that comes as standard with the set, fitted the head directly to the body - and the problem went away.
I've done a full reset on the radio, followed by re-loading the memories from the PC - its working fine in the house.
I suspect that the little end connectors (RJ-45s, or something like that) on the extension cable may have got pulled about a bit as I ran the cables - they are not very robust.
I've ordered a new cable - at under £10, it's just not worth fiddling on any more.
Hopefully, all will be finally sorted next week.
Thanks to all for the help.
Cheers,
Tony M