As much as I love GHS Precision flats for some things, if you want upper mids I don't think these are what you're looking for. They are more full-bodied and have the typical GHS lower mid thing that's more present. Sounds like you might consider some Roto flats or possibly Chromes. I don't think they have much of any mids, but the highs might be helpful. I like the Fender flats (some), better than Chromes--which I don't like I all. The TIs have some good mids and upper mids, but they are either loved or hated. I love them on old Pbasses.
One thing you might try is the GHS Pressurewound "flats." They are not flats, but are advertized as such. They are Pressurewounds and have fantastic mids. I love these strings in many settings. You could consider the EB Cobalts as well. Kind of a different animal but have their own thing going on.