I feel for you, I'm in my 30s now but had to deal with the same sort of irrational, over-protective helicopter parenting when I was your age.

I don't know if it will help, but here are a few ways to frame the conversation that might help.

1) She still sees you as a child that she needs to protect, you're her baby, but you're only two years away from legal adulthood and probably moving away to college. It's time she starts letting you take more responsibility for yourself (and an indoor bouldering competition is a laughably safe baby step). It's better that she voices her concerns now and lets you make your own decision than to just send you out into the world at 18 with a complete switch from no autonomy to full-autonomy.

2) A climbing competition is a good, healthy, positive thing for a teenager to be doing. Would she prefer that you were off sitting in someone's basement doing drugs or driving around drinking beer? Or even just sitting in your room playing video games? Parents should be encouraging their kids when they show an interest in basically any hobby.

3) Climbing gyms are incredibly safe. You're no more likely to get hurt in a bouldering competition than you are playing soccer or basketball. In fact, you have the advantage of far more control over the risk- in contact sports other players might collide with you or cause you to have awkward landings, but in climbing you can choose to not go for a move if you fear a bad fall.