Filling out a false application is a really bad idea because you’re teaching your daughter to lie, and that’s a lesson that could stay with her forever. I’ve no doubt you believe she’s capable of keeping up with the older kids, but you’re not the best judge of that. The obvious answer here is to appeal to the club administrators and let them decide whether they want to make an exception. But be prepared for them to say no (I would) because they may get a lot of such appeals and feel a hard-and-fast rule is best.
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The shortcoming here is that you’ve shifted the ethical issue to someone else, even though if you were in their place you would say “no”.
If you believe the rule is good, which it seems you do, then explain to your daughter why you agree with the rule.
If you believe the rule is bad, or could be improved, then discuss with your daughter to come up with ideas for a better rule. And if you come up with one, present it to the club administrators.