The JB pickups should be fine for what you are looking for. Jake E. Lee used them, Dave Friedman also. Dave Mustaine used a JB for most of the Rust in Peace album as well. I wouldn't worry too much about string gauge either. Correct me if I'm wrong but I think Randy Rhoads used 9's.
As far as amps, the bands you mentioned for the most part use Marshall style amps - JCM 800's and super leads for the most part - Adrian Smith I think was actually using a DSL at one point. The DSL should be able to get you in the ball park of what you are looking for. I don't have any experience with blackstar amps, but have had a DSL and a JCM 800. For me, the 800 was a monster, but had to be cranked to sound right. The DSL sounded ok at bedroom volumes, but cranked up sounded really good - the sweet spot for me seemed to be about 4.5 - 5 on the master volume.
If I was in your shoes with your setup and looking for those tones my first thought would be to try a tube screamer type od in front instead of the sd1. MXR makes a nice one that's cheap as well - the classic od. has an internal switch that can change it from the standard ts type to their zack wylde circuit - can find some info here
https://reverb.com/p/mxr-classic-overdrive
also as suggested earlier, an eq pedal could help. I use a Carl Martin Boost Kick as an eq (even though it's really a boost pedal) and it's not too expensive but in that same price range mxr has a 10 band eq that's pretty nice as well - or for a smaller footprint maybe the boss ge-7.
Those bands you mentioned have pretty "mid" heavy sounds. If you can, I'd try turning the master volume up first (that can make a huge difference in tone on a tube amp like the DSL), If not, without knowing your settings I would guess that you may need to bring your mids up a little and the bass down a little, but that is worth a shot. After that, if it's still needed, I'd look at switching to a tubescreamer type pedal instead of the sd-1 or/and an eq pedal. Adding some mids will definitely help with tightening up the sound
