V.W.
I think it depends on your husband's job. I am a SAHM and my husband NEVER got up with either of ours. He is an air traffic controller, and commutes almost an hour to work. So on top of his high stress and NO room for mistakes job, he has 2 hours of driving to stay awake. Him getting up was never an option. He didn't even get up to "bring me the baby" when I was nursing them. I just got up, quietly flicked off the monitor and went down the hall to feed them.
Fortunately for me, both our kids were good sleepers and were sleeping "through the night" (5 hours at a stretch or more) by 3 months. They did go through those phases later where they show up at the side of the bed several times a night.. but those passed quickly too, with consistency on my part.
We also had NO family nearby to help.
I just tried not to let the other SAHM "duties" get to me. I kept enough laundry done that we never were without clean clothes (and my husband ALWAYS has helped with the laundry. He even would help by cooking meals sometimes or grilling out), and the house was NEVER spotless... usually it was barely presentable. But, I tried to nap when the kids napped (and they gave up naps early, too.. so that wasn't much), and somehow managed to survive. My husband didn't sleep in another room, either, so there was no "co-sleeping" going on with the kids. To be honest, I liked the way it was most of the time. Frankly, even if he HAD gotten up with the babies, I would still have been awake anyway... At least ONE of us was getting some sleep.
It sounds more like you are just exhausted and resent him having "his weekends". That is a whole other issue besides getting up in the night. Maybe on the weekend he could take BOTH the kids somewhere during the day for a few hours so you can nap? Or agree to get up with them first thing in the morning so you can sleep in? Perhaps he could make Friday and Saturday nights HIS night to put them to bed? You could go to bed early and catch up on some zzzzz's.
Nobody else's "schedule" is necessarily going to work for you and your husband... you're going to have to work out a plan that suits your family. But if you allow yourself to focus on the "unfairness" of it, then you will only make yourself miserable. If you want some time for yourself.. then tell him that's what you need, and SCHEDULE a time on the weekend when he will be totally responsible for the kids while you ___ (take a bubble bath, nap, go to a movie, read... whatever).
Hang in there Mama!