I used to have a brace and played tonnes of sports and never had to use a mouthguard, football (soccer), rounders (baseball), cricket, hockey as well as others. However, with saying that I would advise you to wear one, I could have just been lucky. My Dad once told me of an awful accident he had whilst playing hockey when he had his brace as a boy. It would make sense to wear one, especially one for the upper and lower brace.
You only need one for the upper, you just get the ones made for braces were they are larger and squishier. I don't think you could fit two in your mouth, at least not comfortably.
For basketball I had this mouth guard that had an upper and lower protection but it was all one piece. But good luck in lacrosse!
Kay thanks guys, my coach told me just to wear one for the upper
