J.C.
I have to totally agree with Emily B. Though it's hard, I think being honest is the most important thing. I think you should be proud of your boyfriend, which it sounds like you are. This is a very humbling time for him as well and the fact that he is taking the steps to improve his life and the lives of those around him is definitely praiseworthy. If your family and the ex are going to judge him for doing something that is good and is for the betterment of your family, I think the problem lies with them, not you or your boyfriend. You'll feel better for having it all out there, because the little white lie will just eat at you for the next couple of months. I would just be open and honest up front and leave it at that. And I would be very firm and make it clear that there's really no room for discussion after it's out there. A simple, this is what's going on, this is what we're doing about it and call it over.
As for the situation with the ex and daughter, I really don't have much to offer on that. I don't know the law enough to know what rights the ex has in this situation. If your boyfriend isn't there for his given custody responsibilities does she have to leave her with you? I guess my thought is that kids that age will talk without even knowing they are divulging something that you wish they wouldn't. A simple, "Daddy wasn't there tonight" will prompt the ex to call you to find out what's going on. If your boyfriend isn't in rehab yet, I would suggest that he work that out with his ex before he leaves. And if he is already, is there no way for him to contact her to work it out?
Good luck.