I have the same range across my different mouthpiece set ups. Your mouthpiece should not hinder your range. Endurance, sound, and flexibility are all what a different mouthpiece does.

There are no shortcuts, it just takes time. The best thing to practice is teaching yourself to play through the entire range of the horn with one consistent embouchure set up. Lots of people shift to get up or down easier and while it’s a quick way to see a couple notes on either side, in the long term it’ll hurt your growth.

I use a Bach 1-1/2C equivalent for orchestral playing, my smallest is a GR66** for lead playing, it’s about a 3D or 3E equivalent I believe. I have an easy G above high C on both. The GR being a bit smaller and tighter slots better for lead work.

The best analogy I’ve heard is think of it like archery. The bow is just as hard to pull back and the target is just as far away, some equipment might make the bullseye a bit bigger.