Will the Fender set give enough versatility? I don't just want all thud and thump as I do play some different styles of music.
I would consider the Fender 9050 flats to be "affordable, all-purpose and middle-of-the-road" and have always recommended them to anyone who's just venturing into the world of flats for the first time. Not too bright, not too dark...not too stiff, not too floppy.
If you want the old-school thump, get the La Bella. If you want flats that sound like rounds, get the EB Cobalts. If you want flats that still sound and feel like flats but are versatile enough to be used for different styles of music, then the Fender 9050CL should be on your list of strings to try.