I just want to add my own spin to a couple of things, since you've already gotten some good suggestions:
1) You cannot overestimate the value of having him take 3 days off work to stay home all day alone with all 6 kids. Even if he has to use valuable, coveted vacation time to do it, it is equally valuable for him to get a ~real~ sense of what you do. And he needs to do it for more than one day, he needs to understand the monotony of it, and the physical exhaustion that builds up. And he needs to see how important it is that you DO find joy in the monotony, in the exhaustion, because it is time with your kids, and your kids deserve to be surrounded by love and joy (which is, actually, part of your job, to tamp down any unhappiness so that your kids are happy.... not an easy part of the job, I might add!).
During those three days, it would be really helpful if you took a temp job or something, so that you guys really are able to do an "exchange" (you can volunteer to do office work for a non-profit or something, if you have trouble getting hired by a temp agency or don't want to invest in that sort of relationship for just a 3-day gig). it's helpful if you spend those 3 days in his shoes, or as close as possible, so that you can also recognize that special kind of tired that comes from a long day at the office. It sounds like you've never worked an office job, or at least not in a long time, and it would probably do you some good to get some perspective on his life, too.
of course, I'm assuming here that he works an office job.... if he's working construction, or teaching, or a mechanic, or whatever, then adjust my advice accordingly (volunteer for habitat for humanity, or in a local school, or for a trade school... you see where I'm going with this....).
2) Talk talk talk to him. It sounds like he's feeling trapped and resentful. And he has a right to; it is extremely stressful to be the sole breadwinner, even for brief periods of time, and you need to acknowledge that stress for him. It is also not uncommon for a "regular" job to be monotonous, so it may be that you pointing out that there is monotony in your day seems to him to be dismissing the monotony in his. Let him talk about what's bothering him. Really listen. And then ask him what would help make things easier on him. Give him some power to address his own complaints. For example, you mention his comment about you getting to sleep in. If that's true, then maybe you shouldn't get to sleep in. Maybe it would be helpful if you balance your work days, so that you're both up at 6am, for example, and both of you spend your mornings "getting ready for work," which means showering and dressing, getting breakfast (and both of you can be involved in dressing and feeding kids, for example, because neither of you is "on the clock" yet). Then, it's agreed that when he heads out the door, you're both "on the clock," which means he's responsible for his job, and you're responsible for yours. You need to delineate exactly what your job is: is it just the kids? (and make sure he knows what "just" the kids is: outings, education, skills-building, what you'll be doing all day to accomplish what goals in terms of their development -- don't get out of control here, I'm not saying to micromanage your day or to become one of those crazy moms who can't just have fun with her kids.... I'm just saying, make a point of what you'll be doing that isn't sitting around watching TV and painting your nails all day!) and if it's more than the kids, then what? the kids and the house? what does "and the house" mean? cleaning? cooking? laundry? how often? what are his expectations of what you can accomplish during the day? are they reasonable? are you being reasonable? Really talk about this stuff, listen to him, ask that he listen to you, and write down your ideas so that you both end up on the same page. So much of this is about perception and expectations, so if you can clarify those, man, the battle is almost over.
then, once you really lay out your job description (because, for the record, that's what you're doing here: you have a job, and someone is questioning whether you're performing your job appropriately, so you need to sit down and define what your job is, so as better to assess whether or not you're doing your job... I suggest you talk in these terms b/c this isn't about "staying home with the kids," it's about what your job is, what your contribution to the family welfare is, and he needs to see that in the same light, the same terms, as when his job is evaluated.), you need to perform the duties as laid out, and he needs to leave you alone about any "extras."
with your job descriptions understood and your job performance no longer an issue, I suggest that you address your non-work hours and days: because you're both working during the day and thus both "on the clock" all day, the minute he gets home from work, you're both "off the clock." which means you're both done with your work days and you no longer answer to your job descriptions. which means you tag-team the kids, house, dinner, chores, etc. from that point on. At night and on the weekends, if he's not at his job, then you're not at yours. you're both just a couple of parents working together to do the best you can. Sure, sometimes one of you can make a point of "taking it from here" so that the other one gets a break, if one of you has had a particularly bad or particularly good day. but most of the time, you're a team, and you both recognize that you've both had long days but you're both parents and the day isn't over yet.
I really think that so much resentment about who does more (whether or not there are kids involved, but especially when there's a stay-at-home), is because couples forget to communicate and instead rely on passive aggression, insinuation, and side-comments, all of which lead to simmering resentment. Those aren't fun things. Neither are hard conversations, frankly, but they're a heck of a lot better than the alternative, right?
sheesh! this is long. I hope it's helpful, though. good luck!
best,
K.