I use Fender 9050CL flats for everything. All the others will lose that twang after a few gigs, but the Fenders will round out and stabilize with a good tonal signature.
Now, think about this: many of the "classic country" hits of the last forty years were recorded on something like a Precision with the old Fender flats, which were a lot more mellow than the new ones.
That said, have you tried GHS Pressurewounds? Bright without zing.
Have you tried a different gauge? Heavier rounds will have less zing, lighter flats will have more zing, for any given brand or model.