I think the secrets to fast attack are:
– tight gaskets
– a short, stiff mouthpiece.
I’m surprised at how leaky my gaskets are! Just blow into it with no keys down: ideally, you shouldn’t be able to!
And I love the hose, but its flexibility take some of the attack power out of it.
Practice is probably the best diaphragm strengthening exercise, but I bet there are others.
Physics demands that there be more energy to run those low notes, and there’s not much you can do about them. I don’t think you can do anything to the reeds outside of filing them down, sharpening and flattening, to make them thinner and yet stay in tune. That’s precarious.