Unacceptable and unreasonable (and immature) behavior from his boss. She's obviously a control-freak but is way overstepping. Your husband needs to put his foot down. First comes the understanding on his part that family comes *first* absolutely and utmost no "ifs, ands, or buts"
As far as "managing" his boss, if he's scared of her or of the outcome, things will never change for the good. As others said, look for another job so he has something in the back-pocket. Obviously, he can't be all that picky about it but sounds like his skill set commands a competitive salary negotiation. He should try to time hopefully getting the offer so that it coincides with him talking to his boss' boss (is he on good terms with anyone above his boss?). If they can't work something out at the current place, then peace out with your back-pocket offer(s). The sooner he starts this process, the sooner the problem will be dealt with (rather than waiting several weeks to months for something to change). On top of that, even if he does bring it up as a problem with the intention of staying, it's likely going to result in more passive-aggressiveness from that extreme micro-managing supervisor of his (even if say he gets promoted and they become peers - she'll likely be all hostile towards him and try to sabotage him in other ways). As what others have been pointing out already, it seems like moving on might be the best option. Remember, it's for the sake of your family and that's the number one priority. It seems you both are trying to maintain this perspective and it's probably a lot harder for the husband, as the "bread-winner" to forfeit the means of income for his family. But it doesn't mean quitting flat out - he should have enough confidence in himself now to say "screw it, I'm looking elsewhere" and really commit to that direction without fear or anxiety over what his supervisor thinks (sounds like she's really gotten into his head). If he's concerned about "burning bridges," she's already done all that IMHO.