Now that you know he's not going to do anyting to fix the situation, I would be giving corrections. It is *never* acceptable for a dog to bite. A correction or time out is going to have to be your responsibility. If he won't react, you have to. Especially since you have kids. What if it had been a kid instead of you that was bitten? The dog needs to understand here and now that biting is not a behavior that will be tolerated. It doesn't matter if it's never happened before... and chances are, it HAS happened before... you all have only had the dog for 3 months. What happened when the dog was with his ex is anyones guess. Being obnoxious is one thing, biting is another thing entirely, and needs to be treated seriously. He doesn't seem to see it as serious, so I wouldn't go running to him if something happens. And, technically despite it not being your idea, or something you even want, the dog is now both of yours.
Once you have dealt with the situation (if it were to happen again), I would calmly go to him and tell him what happened, what you did to correct the situation, explain why the correction was given, and why it is unacceptable, and I would bring up concerns over the dog grabbing a child. The dog biting is just further proof that rules need to be put in place, and carried out for both dogs. Things are clearly escalating because the dog is allowed to get away with anything it pleases.
Frankly, you are a better person than I. The little brat would either be straightened up and learn the rules of the household with both people making sure rules are carried out, or the dog would be gone. I will not live with a dog like that. Despite me having ten dogs (9 shelties and a BC), none of mine are allowed to behave like that. It's not that they've all started as perfect angels either. The BC came from what I found out later to be a line known for aggressive behavior and dominant behavior, and another two of mine, in the wrong hands, could be fairly dominant dogs, and walk all over someone. However, rules were put in place from the get go, and any unacceptable behavior is corrected as I will not live with a dog who behaves badly.
.... To be honest, things were improving a bit a few weeks ago, but the old girl is a bit arthritic and ended up injuring herself chasing after Miko, so now she even MORE cranky than she was before! One step forward... 9 steps back!
I yell "HEY" and proceed to call the dog every name in the book and tell SO "Your &^&# dog bit me!!!!".... He doesn't move or react in any way shape or form... continues to watch tv. So I went into the washroom to run my finger under cold water (no broken skin, but still teeth marks an hour later) and I end up bursting into tears. I've never been bitten by a dog in my life!! Anyway, FINALLY after about 10 mins SO gets up and comes to see if I'm ok. I burst into tears again... still shaking from the shock of being bitten... and tell him that no, I'm not ok. Anyway... long story short... I get the impression he thinks I'm being dramatic... he's insisting that she's never bitten anyone before (I'm sure he thinks I must have done something to her!)... I assure you... I didn't do anything!! Miko was at least a foot away from her.. nothing startled her... nothing touched her. It was completely unprovoked!! I don't know what to do now... I'm think I'm more hurt by SO's attitude than I am about the d*mn dog! I don't care that she's never bitten anyone before... she bit ME today!!! 
