Honestly I think you need a long hard look at your parenting and work out whether there are any other areas where you're babying your DD. You aren't doing her any favours
The OP has asked a question about suncream, not about her parenting.
My ds is 13 and I put suncream on him just last week. He didn't moan. Admittedly it was at home, not on a beach in public, and only really on the back of his neck and as I had so much on my hands, I spread it onto his arms and shoulders too.
I find it a bit odd that everyone is saying "well she has to make her own mistakes". So if she gets really bad sunburn that's ok because the OP has to let her make her own mistakes? We'll use that principle for road safety too shall we?
People on here have to turn every debate into virtue signalling about how great their parenting is. No parent is perfect but a lot of MNers simply can't grasp that.
OP, get her to do the bits she can do easily and then help her with the rest eg back and neck. And even better - stay in the shade, keep light clothes on, wear a hat and sunglasses and you'll need less of the stuff. I don't understand why you'd want to fry yourself on a beach anyway.