I'm late to the game here, but I've got a lot to say that's been bouncing around my head since I started seeing you post, but I haven't had time to reply because I've been too busy taking care of my baby (my wife and I have our first 4 month old and he's been running us ragged).
Here's my theory: your husband already knows you're right. He knows he should be doing a more fair share of the work, but he hopes that through his various bullshit manbaby tactics he can just make this whole problem go away when you resign yourself to taking on an unreasonable amount of baby care. It may be your best best to let him know that this problem ISN'T just going to go away, the question is only how much damage he wants to do to your partnership before he finally gives in and acts like an adult.
As an aside, at times I think it can be very tempting for each gender to simply accept the traditional roles society has thrust upon us when it benefits them. We know if we examine them too closely we'll realize it's not necessarily fair and work harder at a more egalitarian distribution of work, but I think he's hoping he can just play aloof long enough that, as the woman, you take on all of the child care. But *he needs to grow up and do the manliest thing possible -- embrace, or at least accept, his damn duty to his family. *
For comparison, my wife is also a stay-at-home mom. I work an important job at a major, publicly traded company and earn enough that we live very comfortably. Does that mean that my wife is simply in charge of the baby twenty four hours a day, every day? No, because that would be ridiculous. I work 50-60 hours a week. Thus, it seems fair that my wife should have 50-60 more hours a week of child care than I do. That leaves over 100 hours of week during which we ought to be sharing the baby evenly.
What does that look like? * Every night when I get home I put the baby down for bed so my wife can get an hour to herself. She's the greatest, so she'll often use it to cook us both dinner, but she doesn't have to * I'm on baby duty for a large part of the night, typically til about 3 AM. This way, when my wife goes to bed at 9, she can still get a realistic block of sleep. I can go to bed at 11 and get patchy sleep til 3, then a solid block til 7. Oh, and does my wife hear the baby before I do? Yea, she does, especially because my beautiful wife snores at the volume of a shout and I wear earplugs. But why in the world does that mean the baby is therefor her responsibility at that moment? She shakes me awake and I go take care of business and put the damn baby back to sleep. Yea, my job is important, but so is hers. She's raising our son, and unlike my job, we can't get a new one if this one turns out shitty. * There are baby related chores that are just mine. I wash the bottles every day. I also try and make sure I spent at least 20 minutes a day doing something extra for her that she didn't expect, because it makes such a difference.
This isn't to say that everything needs to be split 50/50 down the middle. It isn't always 50/50 in relationships, you just want it to kind of average out that way over time. Try and focus on each taking the work that you mind the least. But really, do whatever it takes to get this guy to stop being a manbaby and accept his duty to his family, it's making me crazy just knowing that bums like this skate by out there.