Yeah I mean just running a TS-style overdrive (not something transparent, but that has that high-pass and upper-mids thing) will get you there with just about any pickup. If you're in Europe, I'd say check out the Bare Knuckle lineup - the flagship stuff is pricey but really exceptional (not my favorite vibe personally, but the quality is there for sure), and their Boot Camp line is decently affordable - around £170, however that converts to eurobucks - and sounds really good. Rabea Massad did the demos for them and has an overview on his channel, and they're pretty spectacular.
The issue with the kinda flat timbre you're getting is because with kinda bright but kinda nasally pickups (like the stock Sterling pickups in the JPs I've tried), and the massaging you need to do to get them working well for you, you lose a lot of character that the pickups can't necessarily spare. For something tuned that low, you need to have a very tactical approach to managing frequencies (my personal pick is a bright pickup without a lot of bass, thick strings, and a very balanced amp that can go either dark or bright and gives you the flexibility to dial in your tone just right). My $0.02, if you're getting usable sounds out of the stock pickups, dig around for samples and clips of a bunch of different things - including pickups you might not necessarily expect to be what you're after - get a good grasp of the qualities and characteristics that would work well for your specific sound, playing styles, and music, and even if you have to wait to pick something up, you'll get a better end result.
Speaking of - what's your rig like, and what kind of riffs & tones are you looking to get? There's a lot of variety out there for very low-tuned stuff.