to give her away in marriage, although of course it's traditional that her father does the job.
Your situation is a tricky one, and it's difficult for an ouitsider to offer anything but the most basic advice. I suggest that you have a private and tactful discussion with your future daughter-in-law. Explain to her that although you're honoured by her request it might be best for all concerned if she could bring herself to ask her own father to carry out what is, after all, a very quick duty. It will be over with in a flash, and might make him very happy. Otherwise she risks hurting him, and there's no going back. Her day will be a happy one in any event, and by making this one small gesture for the sake of her father's feelings she may find she feels a whole lot better in the long run.
At all costs I caution you against becoming the filling in a family dispute sandwich.
Bursting into tears isn't the worst thing in the world by the way, even when a man does it. Perhaps her father is feeling emotional about the situation between him and his daughter.