We don't know that this was a child though. We do not know how old the person that was bitten is.
Is this someone who lives with you? And, how old is she. If she is 1 or 2 or 3, yeah, I think I agree with your position. If your daughter is from a previous marriage and is 16 or 21, usually lives with her mother, or is away at college, and hasn't had much contact with this dog -- doesn't live there, then I think it is harder for you, because you would be asking your wife to put down her dog, because of something that your daughter did, that probably wasn't a smart move.
If she is an older child, ie. teen or an adult, and does not live with you, then I think you need to understand that while your dog bit, she bit once, caused pain and a wound, but did not maul the daughter. I know it is still awful, especially in the face, but the dog is not crazy, uncontrollable, unpredictable. Maybe she can be safely managed by the person who loves her most -- your wife.
Rehoming a dog that will bite someone like this, is really a question. If your daughter is a child -- aged 10 or under, either work with the dog and an experienced trainer, or take the dog to the vet and have her put down. I think the latter makes more sense, but would be very hard to do.
Sorry you are going through this. Having a dog for 5 years, it would be devastating. Probably moreso for the wife.
BTW, it is the my daughter, not our daughter that makes me think that maybe the daughter is a little older and not the wife's daughter.
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