Why don't you just have her use a Diva Cup? I mean, yes, there's a learning curve, but it's by far better than a pad or tampon. My daughter is very serious about ballet, and although she hasn't started her period yet, she had me buy her a Diva (well, we bought a different brand that is a bit smaller, and came in a purple color instead of just the clear ;).
The reason my daughter is even thinking of this is that she has a friend in her ballet class who had her period during their Summer Intensive program (when they're dancing 8+ hours per day), and for whatever reason was wearing pads instead of a tampon or diva cup. Anyway, this poor girl had a terrible leak that ran all the way down her pink tights to her shoes, and she had to run out of the class (that a male teacher was teaching, eeek!), and change her entire outfit. Let's just say that that experience seems to have scarred that poor girl for life, and she now wears tampons.
Perhaps you could explore with your daughter the reasons she's afraid of tampons or a diva cup, and ask how she feels about a pad inevitably leaking (gymnastics requires as much movement as ballet, if not more!), and what she would do, how she would feel if that happened. Frankly, as a mom, I'd just do what I could to save her from that embarrassment up front. Just my two cents!