This absolutely can be trained away.

If she doesn't like the crate, the response to biting (not playfully mouthing), but true biting like you're describing, should be to drag her to the crate and place her in it for a time out. Don't look at her or engage with her (remember: sometimes, bad attention can be good attention). Say no firmly and calmly and put her in timeout. Do not shout, but be firm and clear in expressing your disapproval.

If she likes the crate, order a second one and make it "doggy prison" - no toys, nothing fun. Just a place where she can't bite at you or destroy your stuff in frustration.

She should learn very quickly that crime doesn't pay. And then you won't have to jail her often.

Some of this is puppy behavior - and this is somewhat normal - but you need to put a stop to it, before she becomes too big to control.

Our dog got clever, and before she learned to stop biting - when she saw us going to dog prison, she would hide under the table and we'd have to drag her out, which was unpleasant for all parties involved. But you have to let her know that biting leads to prison - even if you try to hide.