I seem to be in a cycle of rotating between rounds and flats on my P bass... Rounds on for a while and then the urge to have flatwound old school thump kicks in, so I put flats on. And then I convince myself that the bass sounds great, but would be more usable/"I want the zingyness" with rounds. Then the rounds go dead and I remember how good it sounds with flats, so the cycle repeats. lol I just put rounds back on my P bass last night, actually.
My P bass has had both Chromes and GHS Precision flats on/off twice, and I also had a brief trial of Labella Deep Talkin' flats on a friend's bass. For the latest set of flats, I was right on the fence between Cobalt flats that
@Bald-N-Bassy and
@Kipp Harrington mentioned and Dunlop flats. I decided to go with the Dunlops because I read scattered reports that the Cobalts had a slight feel of stickyness due to how they're wound with a slight bit of spacing in the wraps, and have been very happy with the sound/feel. They're on the modern side of flatwound sounds with a bit of zing when they start out, but play them in and they settle down and balance out nicely. They also seem to have less tension than the Chromes and GHS Precisions, which is nice too.
I've got some audio of the Dunlops if it helps make any decisions one way or the other.
Dunlop Flatwounds freshly installed and
after 7 months of playing.
@drumvsbass - The TI's are on the list for "someday" because of exactly what you mentioned.
Flats obviously need to be changed much less than rounds (if ever), but holy crapspackle the Dunlops have gone WAAAAYYY up in price since I bought mine. I paid $45ish new, and now they're almost double that for the 5-string set.

For the price they're now asking, I would've at least tried the Cobalts which are still that price, but maybe would have just skipped and went directly to the TIs
5sg.