I love ska, my college band was ska, so my opinion matters, blah blah blah. An HH Tele should definitely get you there. Fat, juicy, compressed cleans and raw power distortion is the name of the tone game. I'd think a high gain Marshall model for the dirt parts (with an optional boost) and a Fender model with a compressor and boosted mids for the cleans.
Pickups - honestly a medium output pickup set is going to be your best bet for a solid tonal foundation here. You need the juiciness that low and medium output'ers offer. I just can't see Invaders/Actives giving the right kinds of tones and with a modeler like the Katana you should be able to dial in all the gain you need from the amp and built-in pedal chain. PAFs, wide-range humbuckers, hum-sized P90s should get you the juiciness AND have enough output to slam the amp when you kick in the boost/dirt/OD/dist.
I hope this doesn't come across poorly - just trying to help - but 7 pickup swaps in and you aren't happy, the problem isn't with the pickup, it's somewhere else in the signal chain. Pickups do make a difference, but it's often subtle and if you change nothing else about the signal chain, you're likely to keep ending up being unhappy. Is this with a band? Maybe a single speaker isn't filling out enough sound, and a pickup swap ain't going to fix that, you need more/bigger speakers. If it's home playing, I think you're better served by tweaking your settings more than by another pickup swap. Or even consider a second amplifier for the distorted parts. Tweak the polepieces and pickup height. The trem/non trem spacing generally isn't very significant for volume drop-off between strings, but I haven't played this particular guitar so I don't know for sure. You might need a bigger gauge high E string.
Caveat: a Tele is pretty much always going to sound like a Tele, no matter what you put in it. If the tones you're chasing are made by SGs, Les Pauls, things like that, you are probably not going to satisfied chasing those tones on a Tele. Hard to know sometimes when you're listening to your favorite bands - especially since what the guitars they record with are often different than the guitars they perform with so Youtube and Google Image searches can be misleading.
