I don't have a LP, and none of the ones I had care of in my youth had flats, so I can't comment directly - but when LPs were invented, pretty much all strings were flat-wound (and a little later, when they came out with something as small as .012s, they were dubbed "Light" - still called that by most string companies), so I don't see a problem.
A friend gave me an open-ended loan of a Washburn jazzbox, and the first thing I did was switch it from his round .009s to flat .011s; gave it a little more warmth (from the flatness) and gravitas (from the gauge) is all.
Worst case? The customer doesn't like them, and goes back to rounds.
Peter (who, in truth, would switch the jazzer to .010s, but can't find flat .010s with a wound G)