Op, your husbands stance doesn't make any sense whatsoever. You MUST see that.
If he's an alpha male, and you like that kind of thing (some women do until they see sense), which would be (kind of) fine, in an 'each to their own' fashion.
But, he's not is he? He's picking the bits that suit him (no housework, no childcare); but not that bits that don't, ie you working.
So, therefore, he's not an alpha male, simply a lazy arsehole who has worked out how to manipulate you to his best advantage.
As it stands, if nothing changes, you will end in divorce, without a shadow of a doubt. You will grow more and more resentful amd eventually hate him. Then you'll divorce, find yourself suddenly so happy, and wonder why you didn't do it sooner. Just depends how many years of your one life you want to spend miserable first.
I don't really think he isn't well aware that he should be doing 50% housework and childcare, if you're both working full time. How can he, unless he's thick as mince as well?