That should not be allowed in the US. "Prove that your religious convictions are specifically supported by the people currently in power at your particular church branch" has never been required.

She may need to articulate why her refusal is religious and not just "I don't want to do this," but she should not need an appeal to a higher authority to make that claim.

She legally should not have to articulate the details, but since the next option after "HR refuses the claim" is "take them to court," at some point, she'll probably need to. Doing so in a way that makes it obvious to HR that this is a religious issue for her, is the best way to avoid that court case.