Sorry for the late answer (if it has any interest, which isn't sure). I was ill...
Not easy IMHO and IME to find a "balanced" pickup, since passive magnetic transducers are by nature unbalanced: physically, they are LRC resonant filters. So they have resonant peaks. So they are never flat sounding (unless a tone pot is lowered enough to flatten their resonance, of course).
Low impedance are a notable exception to this rule but almost forgotten now and I've still to hear metal played with a Les Paul Recording...
I see why fellow members have evoked above powerful pickups like the X2N, 500T, D-Activator: they haven't a flat sound BUT they promote the bass range in a way favouring a relatively flat response of fundamental notes from unfretted low E to 24th frets on the high E. Most of the other pickups "naturally" promote high mids more than bass.
I also see the reasons behind the recommendation of Bill Lawrence L500 series and that's potentially the kind of PU that I'd try as well in such a situation : William Lorenz Stich (Bill Lawrence) designed his humbuckers in a different way, making them potentially very versatile if one knows how to wire them and to pair them with tone shaping components. Not that I'm a BL fan: along with various L500 models, I've also here a collection of DiMarzio / Duncan / active pickups. But BL spontaneously came to my mind in this case.
... That said and before to change a pickup, to swap its magnet or to use an external EQ, I'd think twice about its settings (height, angle under the strings) and I'd consider how much passive magnetic pickups are tonally shaped by external factors/components.
Examples of what I mean (explaining why Bill Lawrence was a great advocate of tayloring the sound with external things like pot resistance, cable capacitance and/or added capacitors, Q filters and so on):
https://youtu.be/4BmkaS91NHQ?si=RDYJQz-KaolTLPo6&t=562
https://youtu.be/HJKYIWGl_KI?si=ivg9aOz-t0yK_tOK&t=211
Technical explanations if needed:
http://buildyourguitar.com/resources/lemme/
https://zerocapcable.com/wordpress/?page_id=209
[ignore the commercial discourse about the potentially indifferent no-cap cable at the bottom of the 2d page... but compare the graphs of this page to the response of a really flat sounding pickup like a low impedance one:
https://www.cycfi.com/projects/neo-series/low-impedance/ ]
Non limitative list. HTH.