My child is in his 20s, but when I read your post, I could feel every moment of desperate exhaustion I felt when he was an infant. I am so so sorry you are going through this, and I do think you are doing an amazing job of even putting sentences together, as tired as you are.
I finally did sleep training at 6 months - I agree that 5.5 might be a little too soon especially as he's growing and his food needs are greater than even a month ago. I think "Isn'tthisfun" has some wonderful suggestions about the hind milk - if you are just nursing until he falls asleep, he's not deep-feeding enough and isn't satisfied. I'm intrigued by the eat/awake/sleep order she mentions - I wish I had heard of that. But I think you have to choose one method and stick with it - the changing around is hard on you and baby because no one can get into a routine, and if you are debating whether this method or that will work, you're not going to be confident and resolute about it. I do think rubbing his back until he falls asleep will backfire in the long run though. If he's hungry, he's got to fill up. If he's not, he's got to self-soothe. A quick back pat with no pick-up works if he's in another room, but it doesn't work if he knows you're right there.
I think your husband is off base, frankly. So I'd suggest any of a number of possible solutions, with you choosing what works for you and your relationship. One is to pump, and let him feed the baby at night. Another is to pump and then send him off with the baby and the older child for an entire Saturday, while you go to bed - and I mean to bed. No cleaning the house or doing a quick child-free run to the grocery store. I like the suggestion below to write down every little thing you do during the day. I think maybe your husband is a bit clueless about what goes on during the day - including sunlight and traffic noises and phones ringing, so that "sleep when the baby sleeps" is a little naive when it comes to your body rhythms. You're not going to be rested even if you do squeeze in a 2 hour nap here and there. A third option is to have you go to a hotel for a day and sleep (yes really!), complete with earplugs and eyeshades and maybe one of those white noise machines, with a do-not-disturb sign on the door. And HE can stay home on a work day or on a Saturday with some pumped breast milk and some bottles, and HE can just sleep when the baby does since he thinks it's so easy! What that might create is a little more understanding of what you are going through, and that would get rid of his dismissive attitude so that you can work on a solution together.
Fourth, and this isn't a single solution, just an add-on to one of the above, you should think about is relaxing your standards - absolutely stop cleaning the house! It really doesn't need as much as you think, and he can do the rest of that when he gets home on whatever schedule works for him. Yes, you want to be available to your older child, and that's important - and I completely understand why you feel resentful all the time. So I'd let housework go entirely, so it can be done on the weekends (by hubby) or in the evening (by hubby), since you are doing what he said, which is to sleep with the baby. I also think your husband can get up at 5 AM on Saturday and Sunday, give the baby a bottle of pumped breast milk, and let you sleep in while he keeps the older one occupied with breakfast and play time too. If he thinks it's so easy for you 5 days a week, he can do it 2 days. I also think you might consider the possibility that you aren't producing enough milk at one feeding to fill that baby up - and that's because you are exhausted.
Overall, I think some men just don't "get it", not because they are all mean and selfish, but because they are totally inexperienced in this. He wants to keep the baby in your room for 6 months - fine, but he's not dealing with it all the time because you are the one doing the nursing. I think it's very helpful for them to experience a little of it, to help them understand (or, if necessary, insist on it) what a toll this takes on you. He works long hours I am sure, but then he is DONE at 6 PM! At least he has the option of veging out in the evenings or on the weekends. He needs a night or two of total sleep interruption to really "get it." And then you can be on the same page. If part of the reason you are cleaning the house is that he has standards there, he needs to relax those. If part of the reason you are doing it is to feel in control of something, anything, then it's kind of on you to back off a bit. No one in the universe, no reasonable person anyway, really expects a mom of an infant to have the house all together all the time. Put 2 baskets in the prime room (family room, whatever), and one gets baby stuff and one gets older kid stuff. That's called "cleaning" - period. Your 7 year old and your husband are perfectly capable of sorting through to find what's needed.
Anyway, I hope you find a solution. Yes, it's true that these days don't last forever. But when you're in the middle of them, they absolutely feel like forever! And depression is a serious condition and needs to be addressed in any number of ways. See that your husband helps you do that, and not by talking you out of it or minimizing the work you do all day.
Good luck!