I have some very recent experience I can share. Just today I swapped a set of Fender flats and a set of Fender rounds across 2 Fender P basses. I don't know the part numbers of the strings unfortunately. I did this out of sheer curiosity.
For starters the first bass is a MIM standard Precision, no mods. It had the round wounds on it and sounded good, like a P should.
The second bass is a Highway One Precision, no mods. It had the flats on it and sounded really good, exactly like a P should sound with flats.
I was most interested in the fact that the Highway One has the Badass II bridge and the other has the bent metal bridge, and was wondering what each bass would sound like with the strings swapped. Now that they are swapped each bass sounds like the other one did before the swap. I don't think the high mass bridge makes much of a difference with flats, but I will admit the sample size is small with just the 2 basses.
But I've become a real fan of P's with flats, to the point that a P with rounds sounds funny now.