I'm using flats on everything I play & there's plenty of 'tone range' to be had with different strings.
On Precisions I've got LaBella Deep Talkin' 1954 Originals, Fender 9050M, TI Jazz Flats, GHS Black Nylons & GHS Brite Flats.
The Brite Flats are Groundwound & do exactly what it says on the packet, I think you'd like them
Although they're not common here, Stringbusters do them I believe, usually get mine fron USA ebay, got some GHS Precision Flats on order now.
Cheerz, John