Even the nicest dog will bite if it feels threatened, or is hurt. I work in a dog daycare, and some of the dogs there do bite. Dogs are like people and have different personalities and moods from day to day, and one can't just invade any dog's space without knowing how the dog will react.
Since you said you are paying for the medical bills for this bite, it seems like you are taking responsibility for what happened, to a certain extent. I do think that your girlfriend may have been provoking the dog in some way, whether intentionally or not. Also, why was her face so close to the dog's that he was able to bite her directly on the nose?? Sounds suspicious to me. One of the big rules at my work is don't put your face in a dog's face. That is just asking to be bitten, especially if the dog is already excited/agitated, etc.
Also, assuming that you have had the dog longer than the girlfriend, I personally would have a hard time justifying getting rid of the dog, unless you were will to sacrifice such a thing for her, which it doesn't seem you are. It is not cruel to keep a dog in a crate at night, or when out of the home, but it would be when you are around it because dogs are social creatures and like to hang out with their packs.
Ugh....sounds like a real shitty situation. I would DTMFA.