Have been where you are OP, twice, although my DH was more supportive. Just going to chuck out some ideas.
Breastfeeding means that you will pretty much be doing all the work at night until you stop. Don't underestimate the impact of this. Expressing is knackering and unrealistic unless the baby is used to bottles from the start. After six months I could never express enough (even using the best electric pumps) for even one feed. I am stopping b-feeding DD2 (13 months) for the sake of my mental health and so that DH can share the nightwork, I was on my knees and felt depressed. We have stopped feeds at night first and DH dealt with the first, horrible 4 or 5 nights (whilst on annual leave).
Until you stop feeding, your DP could take the baby to a different room from 6am and your eldest DC could go to DP in the mornings too, so that you can get some sleep then. When the baby is 6 months maybe he could give breakfast to both DC.
After you do stop feeding, your DP could do 3 of 7 nights every week. Don't take any crap about him needing rest or having to work: you have to look after the DC and regain your health.
Does your DP do 50% of the childcare / housework / shopping etc. ? If not, then this is part of the problem and changing it could help you a lot.
Instead of your DP's proposal for you to take 48 hours off (as you say, not possible when b-feeding), he could have both DC for the entire weekend while you remain at home but do what you wish (sleep, have long bath, go for a walk or coffee or to shops locally), and he brings the baby to you when s/he needs to feed. He would do absolutely everything else in terms of childcare, housework, cooking etc (as if you weren't there). Then, after the first time, he could do this for one day, say, every two or three weeks. Your eldest DC could just be told that mummy is tired or ill and resting.
In the evenings in the week (assuming your DP works), you could eat with the DC and DP sort himself out later (unless, of course, you enjoy cooking / eating with your DP or your DP cooks). This is good for energy levels, might cheer you up at a peak argument time (evenings when partner gets in) and reduces washing up etc. in the evenings.
Your DP could learn to drive. Driving while sleep-deprived and breastfeeding is really hard, there're days when I don't as feel it wouldn't be safe. Even if money is a problem, make driving lessons a priority, maybe he has a mate who could help teach him in exchange for something.
Avoid outings as a family that are stressful. If meals out are stressful, save that for times when you have a babysitter, do simple stuff like park, soft play or whatever instead.
If you have anyone who can babysit for a couple of hours in the day, go for lunch or coffee with DP. Better than evening meal out when sleep-deprived.
Do you have mum friends to moan with? I didn't have this with DD1 but do with DD2 and it has made a big difference. Often, even people who are seemingly "everything is great" on the surface will swap tales and complain about things if you open up! Helps to get stuff off your chest. ALL the mums I know with 2 DC under 3 are struggling, and all have good and bad days. The ones who struggle the most are, in general, the ones whose partners do less of the work.
Sorry this is long, just trying to think what might help.