I did once have a letter from the CAA stating that my FAA certificate privileges could be exercised in a G reg aeroplane, including all ratings, without formality. They sent me the section from the ANO and the restrictions placed upon the IR. I'd assume this is good enough to act as a "validation" of the foreign license to show foreign authorities if ramp checked or to show a flight school / syndicate. If you write to them and ask a specific question they normally reply (though saying that they have yet to reply to my last email of 2 months ago - I'm expecting the "sorry for the delay in replying" letter soon
).
FTR one of our syndicate only has an FAA PPL for fixed wing and our current aeroplane is G reg (despite being a famous aviator who flew around the world non stop several years ago). It has never been an issue wrt insurance or flying to France.