I find "lost media" isn't really a fitting description of library music - as long as it exists
somewhere, it's not really lost. And if it's truly lost media, then there is no point in looking for it, as that means it doesn't exist anymore.
"A Frenchman in New York" appears on MTS 2003/2004 as you mention, though under its original French title "Un français a New-York". I unfortunately don't have this LP myself and don't think I've seen a rip of it pop up yet - but have seen many LP copies for sale over the years, it's not that rare. The recording was also reissued on
Master Plus, 38022 "Hollywood Moments" (LP).
There are a few cheap copies of both for sale on Discogs at the moment.
It might be remastered or altered in some minor way, but I'm pretty sure the Bella Musica CD you mention contains the same recording; the date, length and copyright holder all check out. As far as I can tell, this CD is also available on several streaming sites.