I dealt with this as a kid; by the time I had my period I was dancing pretty seriously five days a week. I remember the first time I had my period was during a performance week, and I wore my regular undies the first night, to hold the pad, and kept getting comments from other people about it, and my mom noticed. The next day she went out and bought me a really high-cut thong, something WAY too sexy for a 12 year old
but it did the job nicely-- held the pad while being completely invisible under my costume. Combine that with a thin pad with no wings, and it worked just fine. But I felt miserably self-conscious still, like everybody could see the bulge, and the pad made it harder to move naturally. My mother kind of casually mentioned tampons, and left a box with instructions for me to fiddle with, and that's what I wound up using.
"No undies" rules are common in the dance world-- in some competitions, for example, even for very young girls, visible panty line makes you lose points. Gymnasts and swimmers also have this problem. The ballet world is known already for plenty of other anti-feminist offenses. The more serious ballet schools have gotten plenty of publicity for basically forcing growing girls onto restrictive diets to control their weight, even when the girls were already crazy-skinny.
For the OP-- what about a high-cut thong? That was a good solution for me, until I was a bit more mature and ready to try internal protection. But I also agree that if a girl is interested, and WANTS to try tampons, there's no reason she can't try tampons. Any girl old enough to menstruate is old enough to try one. A cup is also an option, or sponges.