Hello Bo, funny thing about piccolo trumpets is that it’s actually easier to play high notes on a longer tube… so why make piccs? sound timbre is my guess. it’s a very different instrument than a Bb trumpet. So yes your initial idea that you are pushing too much air is generally correct. That said, playing picc takes less air but maybe a little more body compression without overdoing it.
When I am prepping something on picc, I will put it in the middle of my day and play short phrases on it with lots of rest in between (play some Bb, rest, play picc, rest, play Bb). I will continue this pattern of back and forth until picc feels comfortable. start where pitch is familiar. middle C on a Bb is low C on a picc (if you are using the Bb pipe) and slowly go up from there. Eyes closed, BCH and all that good stuff. Be gentle. delicate with it. it’s not the laser beam a trumpet can be. do alot of short sessions with it.
I am not sure what else you want to know…all the other stuff to do on picc is basically the same that Greg talks about here… mouth shape, articulations, range is after all mouth shape, the breathing..over time you will find less is better with a picc.. but that balance is up to you to feel the most comfortable with.
I hope that helps and doesnt further confuse.. I am sure Greg has better input