OP - That he pulled out "time of month" is so idiotic that it would be pretty funny, if it were not such a vulgar way to dismiss women as supposedly less rational than men. Men are just as capable of irrationality as women - we are all primates, not robots. Just try bringing bad news to the male boss the day after his favourite football team has lost an important match.
More seriously though, maybe your DH is genuinely baffled if he is not used to having you challenge his assumptions about what women can or cannot do. Give him a chance. He seems to think of himself as reasonable. Lipstick is assuming that he can only be an irredeemable caveman, but maybe he is just clueless. I was, myself, until some of my brilliant female colleagues helped me to see certain things. If he loves you and is enough parts reasonable, then maybe he will want to listen and will then slowly understand certain things. If he does, it could even make him both happier and more successful.
An assumption that could potentially sink his company seems embedded in what he said about how "it’s hard to find the right skills set, experience or commitment levels at middle management level with women". The way in which "commitment" is used here suggests he may think that only a person who can work long hours and is available 24/7 to the company can fill a middle management role effectively.
Many roles require a degree of flexibility, but a company whose success depends on 24/7 commitment, particularly at the middle management level, is doomed to failure. The most talented and capable men and women for these roles will have options other than his company, and many of them will increasingly insist to have time with their families. In the long-term, if he wants to have good people taking care of his company's business, he will need to change the culture in order to enable that.
If he wants to hire someone who can figure out how structure middle management roles in a way that allows high calibre delivery of the work at the same time as having a life outside of the office, then other things being equal, he's more likely to get that by hiring a woman. Because women usually take on more childcare / domestic responsibilities than men, I expect that more women than men have figured out how to manage "work-life balance", if only by necessity. If my hypothesis is correct, then if he hires a woman instead of a man, he's more likely to get someone who has figured out how to organise and prioritise their work effectively and avoid wasting their time at work. He's then also more likely to get someone who can help others to do the same.
If he gets a person (who may well be a woman) who brings a different assumption set about how to do the work, who is more willing to ask "why do we still need to do X", maybe he will even find it easier, himself, to identify things he can eliminate from his own stack of work, and thus free up more time for activities outside of his firm, which you said he personally wants to do.
Note - in order for this to work he would probably need to find a way to hire (at least) two women into senior roles - having just one woman in a roomful of men is a difficult spot to be in, especially given that men are more likely to interrupt / cut off / shout down anyone with whom they disagree. He would probably also need to change how he listens: the number of times I've heard a woman's point adopted by the group and attributed to the man who repeated it is embarrassing. He would need to show leadership by calling that out (e.g. "Actually Fred, that was Gina's point, and I'm glad you agree that it was an excellent point. Thank you for expanding on it.")