N.W.
I've been teaching gymnastics to 3-year olds for 19 years, and what you're experiencing is not uncommon at all! Especially for that age.
Many 3-year olds won't admit they're scared, or don't really know what's going on so they act snotty on the outside. Don't take the snotty attitude as being rude. She is just unsure and that's the way she chooses to display that emotion.
If she's a young 3, then my suggestion is try a parent-tot class. We often put 3-year olds in a parent-tot when they refuse to participate in the regular 3-year old class. It's OK, even if the class is younger (typically it's 18 months to 3 years for our parent-tot) and parent-tot classes help them get used to the classroom setting WITH a parent. Usually it only takes one session and then they're off and running on their own!
If a parent-tot class is not an option, then talk to the instructor. He or she should know how to handle it best with the style they teach. I usually let the parent come in and sit with the child until they get comfortable. Not all instructors can do that. Some of the 3-year-olds sit on the sidelines for 2-3 classes just watching. Then suddenly they join in like they've been there all along! The trick is not to push it too much (a little pushing is necessary). And also, know your child's limits. Some children love to put on a drama show of tears, then as soon as the door shuts they magically turn off the faucet and have a blast.
Otherwise, just give it some time. There is a WORLD of difference between a 3-year old and a 4-year old, believe me! Don't waste your money, try again next year. In the meantime, take her to the pool, or put on some music at home and let her free dance. No need to rush the structured classes. As long as she's moving and learning to love the activity on her terms, there's plenty of time for structured learning at age 4 or 5!