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Peter, I am genuinely mystified by this view you have that FAA IFR approved GPS (with GPS approaches in the AFM Supplement) fits are incredibly rare in the UK.
Mine was done about 3 years ago so I don't remember all the details, but the process is straightforward enough, if a little involved.
There are 2 steps
1. the installation of the equipment
Your engineering people do the installation and an IA signs the Form 337. I was having a lot of work done and moving it to the N-reg at the time, so also needed a DER and DAR to sign things off, but I'm not sure an IFR GPS would need more than a 337 and an IA. I think the 337 is sent to the London IFO, and unless they have an issue with it, it's filed and you are fine.
At this point the GPS is placarded "VFR only".
2. IFR approval
This needs two things: a flight test report in accordance with AC20-138A and a Flight Manual Supplement.
You can do the former yourself, and the latter the engineering people get from Garmin and tweak if required. They send both to the London IFO, and you get the FM Supp back signed.
That's it.
I found no problems in any step of this process and it shouldn't take more than a few weeks. My avionics people have done it several times since.
Unless something has changed, if an avionics company tells you it's impossible or can't be done or very very hard then (IMHO) it's because they don't know how to. The above description should be about right, but I wasn't closely involved except for the flight test report, so I may have got something muddled.
rgds
421C