Yes it was a mistake to allow her to play in the crate. You know better now. i would not assume that you missed any kind of warning sign... but you should understand what went on. It's not a matter of "trusting him enough" to allow her to do that. It's a matter of understanding the dog NEEDS a place that is his, and that is a "safe" place for him to go.
Another element to this is hierarchy. Dogs are pack animals, and you guys (and your daughter) are his pack. There is an alpha male and an alpha female, and then everyone else... who also has their OWN pecking order. Your daughter needs to be above the dog. If the dog was pulling her out of the crate, then he is higher on the pecking order than she is. If he understood himself to be subordinate to her, he wouldn't have done that.
It shouldn't have been allowed to happen (the situation in which she was in his safe place)... but it has, and now he has shown (however politely) that he is dominant to her.
Do not eve allow her to play in the crate. The crate is for the dog to get away and be alone if he needs or wants to. A safe place, from your daughter even, if she is annoying him. Make it a simple rule never to be broken.
And keep a very close eye on them, because litter mates will challenge each other for pack hierarchy.
I will tell you, it is very difficult to teach the child to be dominant over the dog. The dog naturally will dominate... kids just aren't mature enough to possess or impart that "alpha" dominance over the dog. Even their voices are not naturally deep and dominant, but helpless sounding.
Keep your eyes/ears open for any other places where litter mates would need to behave in a dominant manner and do not let the dog become the dominant one. Food and feeding time especially. Or with toys. Stealing (yes, dogs steal) is a big one... the dog will swipe a plaything of your daughter's and take off with it for what looks like a fun (but annoying) game of catch-me-if-you-can... it is stealing. And it is not something to allow to go on.
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But otherwise, the dog sounds very gentle and loving towards your daughter. Our dog grew up with our kids also (daughter was about to turn 3 when we got our puppy) and she is VERY gentle with her mouth. She has put her mouth on kids who were bothering her (my kids) but never bitten. Believe me, there is a major difference. She has never bitten a soul. But she lets you know by putting her mouth on you (with no pressure) to stop whatever it is, that she doesn't like it. If you continue, she will submit to whatever it is, or leave (like when I clip her nails... she sometimes gets tired of it and puts her mouth on my hand... but I push her mouth away and tell her no, and continue, and she lies back and lets me complete the clipping, or manages to get up and walk away, lol--she's big, so other than by my commands, I can't make her stay lying down while I clip without a 2nd person to assist).
But the kids also were taught from an early age that when she mouths them to stop whatever it is... it is a doggy warning! When we play/roughhouse with the kids ourselves, she gets in on it, too.. and will pull on their clothes (jeans leg or shirt sleeve) with no skin or flesh involvement whatsoever. She is very sensitive to what is in her mouth. It's amazing, really.
She is the same when she plucks her frisbee from the bushes outside... you'd think it was a tiny, delicate crystal figurine she was ever so gently removing from the branches of an azalea bush.. but nope: her rubbery, plastic dog frisbee from Old Navy. She's just careful. :)
Sounds like your dog is careful, too. But, because of the nature of a young child and a young, large, gangly dog... you need to be watchful, aware, and careful yourself.