Let me describe how it went for me today - it may help you. The most fear they felt, I think, was when I was catching them up and they tried to get away. Yours won't, I know - but I say this so you know these birds were somewhat fearful of me already.
For each one, I carried him in my arms, talked to him and stroked him. He was alert but calm and peaceful. There was no struggle or fear. As I held him upside down, he flapped for a moment to get his bearings, and then calmed again. I bent over to place his breast and head on the ground, and he did not resist nor try to get away; he trusted me. I held his legs up, and he held himself still with head on the ground. Still no fear. I placed the broomstick across his neck just behind the head (he's still calm and quiet), and quickly put my feet on the stick on either side, and just as quickly pulled up on his feet until the neck snapped.
I think it's not so much your girl's fear that bothers you; it's your own. That's okay and I understand. She will be okay. She won't feel a thing. And with you or your husband - the two people she knows best - holding her and talking to her before the deed, she won't even suspect what's coming.