The shop may actually have smaller mouthpieces. I know I do for when I have small women or youngsters for whom the ordinary one is too big--just ask them about it. It's absolutely trivial to change these, and it doesn't represent non-standard equipment or anything that has to wait until you buy your own gear. If that solves the problem, it's very cheap to buy your own--get a few zip ties, and every time you rent gear, just switch out the mouthpiece on the rental for your own, and you're golden!
