OK, it seems to me that your child is likely to be anxious for a short time while she's waiting to be picked up (which is what happens when you're there). If your backup person is someone she knows and trusts, it seems to me that she's as safe as if you were there.
Maybe talk to your daughter and ask how she would feel about the backup plan? Not that you should put it in terms of her making the decision, but ask her how that would feel for her--would she feel OK if you were away and auntie so-and-so had to pick her up if she called? I mean, if she DID call and need to be picked up, you could be back in a matter of hours, it sounds like. So you could be there shortly to provide any additional comforting she needed.
Obviously you know your daughter better than I do. So maybe her anxiety would be too great. Still, it's good to learn that anxiety can be managed. If you've made provisions for her to leave if things get icky with dad, she should learn to manage a brief period of discomfort as long as she's safe.