This has nothing to do with him being a SAHP and everything to do with him being an ungrateful, lazy, selfish ass.
The problem is that there ARE some options you could explore to make things easier for both of you, but they only work if he is on board. and based on what you're saying, he doesn't sound like he would be onboard.
So, for example, when DH was a SAHP, we both agreed that we would nonetheless have some childcare. I knew that I couldn't be a full time SAHP without wanting to scream, so I didn't begrudge him that. What that looked like varied at different times but it always amounted to the equivalent of one full day a week. That was his day for chilling, relaxing, going to appointments, catching up on admin etc. He didn't usually do excessive cleaning/tidying or whatever, but he definitely did all the bits that were hard to do with a baby in tow and it was when he got his hair cut, saw the dentist, caught up with friends etc.
This also meant he was generally uite happy to let me have some time on the weekend - obviously i wanted to see the DC but frankly, I also needed downtime with a hectic job.
We DID have a cleaner, weekly in those days. That was really important to me. He was vey happy and able o keep things generlaly clean and tidy, do laundry etc, but a cleaner removed the crisis questions of whether floors had been vacuumed/mopped or the bathroom scrubbed.
Dh has the planning skills of a new born baby so somewhat begrudgingly, I took on the effort to identify and plan activities for him and the DC, or sometimes to make suggestions about what they could do during the day. But once they were in place, he would stick to the routine and tended to engage diretctly with the classes etc. It did mean they went to the same 3 softplays a LOT, for example, because he just didn't have the ability to think and plan and research, but quite frnakly, that is a long-standing issue with him and why at the age of 50 he's actually considering a diagnosis for ADHD and meds!!! Plus, the DC didn't mind - they picked GREAT softplays and classes to go to on repeat! 
I kept the cooking (see above re DH's skills) but dropped ALL other basic food prep, or cleaning for years. I didn't load a dishwasher or prepare breakfast or make a lunch box. I also did meal planning and online shopping - delivered while Dh was at home to sort.
The point was that while he wasn't a "natural" SAHD, and I DID help him, he stepped up. He did night wakings. He kept on top of things. The effort I put in he appreciated and reciprocated. I remember sometimes he'd text me when he knew I'd be on the train home and say, "stop at the pub or something - DS is almost asleep and if you come in now he'll want you and you'll be stuck there for hours" then he'd text me the all clear 20 minutes later! The house was broadly tidy unless they'd had a crazy day. I didn't have to think about laundry. We shared lie ins.... basically, we acted as a partnership. it wasn't always smooth sailing but broadly, we were travelling in the same direction.