You work for the company that runs this gymastics program.
That's a good reason NOT to have your child involved on a team run by your employer.
I know you say, "I think it looks bad to work for one company and send my daughter to another" but to whom does it look bad? Other parents? Your employer? I would think you could have a good opportunity here to say: "I love this company so much that heaven forbid I should cause any appearance of a conflict of interest, so to protect the company from any parent claiming that I favor my child over customers' children, I send her elsewhere."
If the company had an issue with that -- well, I'm not sure that I for one would want to work for an employer that had an issue with it once you'd explained it in that way. Nor would I want to work for a company that would expect me to send my child there, and there only, or a company that would penalize me as a worker for not sending my child there. Do you want to work for an employer like that?
Sure, some parent sometime will say, "What, your own employer's company isn't good enough for you?" But you can easily reply, "It's great. I'd love to have my kid here. But I coach here and feel that it's best for the company that there not be any opening for anyone to say I favor my kid."
The other potential solution: Your daughter stays put, but you stop coaching the team. Must you coach it? Or do you choose to do so? Is coaching this specific team part of your job requirements? If it is not -- I'd stop coaching any team my child was on.
Why must she quit altogether rather than just move to another gymnastics program? Don't let some idea of "she must go to the program at the place where I work" prevent her from doing something she loves. You also mentioned cost, but if this is really her passion, can you find the money or work out a payment plan with another place?
If the alternatives for her are no gymnastics at all or gymnastics ONLY at mom's gym -- that's no real alternative. Research alternatives elsewhere--where she can meet new kids who do not know her as the coach's child. She's getting teased because she's good, sure, and that could happen elsewhere, but you have to realize that the fact she's a coach's kid makes it all much worse for her.
And don't offer to coach or even help at the new place -- let it be hers and hers alone.