Flatwound, low tension, through body, bright tone; need recommendations.

Sadowsky Black light gauge flats meet your criteria. I had them strung through-body on a a bass back when I believed that stringing through body actually made any type of difference (other than drastically narrowing my string choices).
 
The TIs or the Cobalts, The TI's are a bit pricey, but have the softest feel. To my ears, the Cobalts sound like perfectly broken in rounds (in terms of brightness), without the round round fret noise. Either will last a good long time.
 
Curt Mangan flatwounds? Not sure if those would meet your criteria.

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45-105 Stainless FlatWound Bass Guitar String Set
 
Cobalt Flats are some of the brightest I've ever played with low tension, and I used to string through with my Lakland 44-51. TI flats are a great option as well, especially if you love low tension. Chromes start off relatively bright and have multiple gauge sets. I personally like Dunlop flats too, very wide tonal spectrum. Good luck!
 
Two Recs:

Ernie Ball Slinky Flats: As others have mentioned, these are the brightest flats, and not have medium-low tension. I have the hybrid 45-100 set, and I can't really imagine needing lower tension.

Dunlop Flats 40-100 have a really good feel on my string-through Mustang, and could possibly be even better with a full-size instrument. They age into a more classic flatwound sound, but they always have some high end and high-mid, and I imagine you could get plenty more with a Stingray.
 
D'Addario flats on a Rick 4003 would work if you want a through-body and can swap out stock rounds. This might work, but I'm happy with the rounds.
If it has to be a through, maybe a StingRay? How does the MM look and feel to you?

The tone you're looking for could be D'Addario flats on a Jazz or Precision, but it depends on the basses you own and specific tone you're looking for.
 
They are? I have to turn the tone on my Jazz bass way down... and the bridge pup too... otherwise it's too bright for me.

You have to do that with most flat wound strings. With that said GHS Precision Flats are probably one of the most "flattest" sounding strings out there. Just do a search here. String noise and fret noise are much different that brightness. I dial back the tone knob on every set of flats I have to get rid of the fretting and string noise.