I have often tried adding a little dirt via a Fulltone Bass Driver or playing palm muted with rounds in the past, only to find that it wouldn't blend with any of the music I was playing.
I've got flats on my bass now, and on a recent blues recording session used both of those sounds. Not typical blues sounds, but we were trying to stretch out a bit.
I think the less aggressive nature of flats made the slight edge of pick and/or overdrive work really well when we wanted to push it a bit. Then a clean tone worked really nicely for a more typical shuffle with acoustic guitars. Articulate without being bright or aggressive at all.
Just something to think about.
It certainly takes a bit of time to get used to flats. Somewhat higher tension, significantly different feeling on the fingers for both hands, a little more burn (for me) on the left hand fingers when sliding around on the fret board. That being said, I've only had them on for about two weeks and feel pretty comfortable with them now.
I should add that I considered going with the light 45-100 set (which would be one step down from the medium rounds I was using before) but according to the tension measurements that D'Addario so graciously puts on the back of their string set boxes, the tension of the medium flats is just barely higher than the medium rounds, where the light rounds are significantly lower. I'm sure that's not the only thing that contributes to the "perceived" tension but, for me, it was what settled the decision to go with the medium Chromes, and I'm glad I did.