I'd suggest the LaBella Deep Talking Bass White Tape Wound strings. They may not technically be called flatwounds because they are wound with tape, but they are smoother than flat wounds and the White tapes are the brightest sounding of their taped offerings. The black tapes are the deepest, the white over copper core and white over gold core are brighter than the black, and the white has the brightest sound of the bunch.
I bought a set over 5-years ago now and swear by them. I originally used flatwound strings on my 1965 Jazz bass because that's what it came with and there really were no round wounds to be found at the time. I actually broke a G string while gigging with it and replaced the set with the Fender Black Nylon Tape Wound strings, which were darker sounding than the regular flats. I used those from about 1968 until 1976 when my '65 Jazz was stolen.
In 1985 I found another Jazz that had the same neck shape, profile, and radius as my '65. It's a 1982 Jazz that was in mint condition and it had Dean Markley Blue Steel roundwound strings on it. That bass also has an all-maple neck and uses the 70's era pickup spacing, so it's a bit brighter sounding bass than my '65 was to start with. Replaced those Blue Steel strings with the same thing at least two or three times and then went with the DR Black Beauty round wounds at a friend's recommendation. They had a better balanced sound from string to string than the Blue Steel did. I gigged a lot with those and turned them into Speckled "Gray Beauty" strings with bright steel bars on the back of the strings where they meet the frets.
Saw a video on SBL (Scott's Bass Lessons) where he was putting rounds, flats, and tapes on the same bass and playing the same musical bit with each. He was a bit enamored with the tapes and so was I. So I bought a set of them from the TB store here and went with white because I have a white bass.
They still have most of the high harmonics I got from the other strings, with better mids and very clear (not muddy) lows. The balance of sound string to string is superb. I no longer feel I have to shift my hand up to the 7th fret to play patterns with a high E or a high A. I can play those notes on the D string down by the head and the G string down by the head with plenty of body to the sound. Playing those notes on the round wounds and even my old Fender flats always sounded a bit weak, thin, and maybe even tinny with the round wounds. Not with these LaBella Tapes though. I can play those notes either up high on the neck or down low on it, whatever is the most convenient hand position for what I am already playing and it sounds great... so I do.
You should give them a try. I think you'd like them.