For what you’re describing I would go with:
GHS Pressure Wound/Ken Smith Compressors
GHS Pure Nickel
DR Sunbeams or Pure Blues
TI flats
The TIs can be very growly, especially on a Jazz and they feature prominent mids.
The Sunbeams and Pure Blues are similar but the Sunbeams are more scooped and lower tension whereas the Pure Blues are probably a little closer to what you want.
I’m fairly new to the Pure Nickels but they’re warm and punchy and I use them on my Kawai FIIB to tame its insane brightness. They sound great on that bass and might sound great on a Jazz, too, although they might be even warmer than you want.
The Compressors/Pressure Wound are more or less the same string in slightly different gauges and they deliver exactly what you describe in terms of tone. The tension is a little on the higher side but not bad. These are some of the few strings left on the market using an iron/nickel alloy and that mid punch is a trademark of that alloy. If you can track down Sfarzo Signature strings, they use the same alloy and they kick serious butt on a Jazz but their distribution is practically non-existent.
I’ve never used SIT Silencers but everything I’ve read here suggests that they’d be a good choice for you, too.
Now you’ll just have to ride the trial-and-error bus for a while until you find the right match.