OP I was up and down with DC3 until he was 2 1/2 years old. I am sure I died a couple of times and came back to life. It is brutal.
I think because this wasn't our first child, it was so much easier for DH and I to sit down and just 'own' that we were deep in the trenches. Our sex life took an enormous hit and I just co-slept in another room with our son. It was tough and very drawn out (which is not necessarily the case for you. I co-slept with my other kids but by 8-12 months old, they were sorted, sleep-wise, especially DC2).
I think if you and DH can sit down and make a pact that you're both in the trenches, that in itself will help ease pressure off the pair of you. It's a real 'get through it' phase. Go to bed earlier. So boring. But do it. It won't be forever. Know that the 5am wake up calls are a phase. Take the 7-9 offer and go to sleep for those two hours while DH takes over. You can fight over this or go with it. He does work all day, he does help out around the house, and it's only fair to take into account that night wakings will interfere with his performance at work. You do have the option of napping with your little one during the day, I assume (unless you have other kids and this is really impossible to do if that's the case).
Choose your battles. Believe me, I feel for you. I could have brained DH at times for his lack of support. He was rubbish at times. I felt so resentful. But once we sat down and said, "This sucks. We love our DC but f*ck this is so hard. I need to co-sleep and just catch any bit of sleep I can, when I can," it was like a weight lifted. We both agreed that we were too knackered to even deal with each other, so we gave each other space and support. I did the lion's share with the kids and took the 'lack of sleep' blow. But on weekends, he would take over and I would sleep in. Sometimes, he would just say, 'Go to bed. Lie down. Get some space. You're getting grizzly.' And I would do it.
I felt unhinged! Lack of sleep is just awful.