First of all, I think we ALL have to get out of the habit of calling it "helping" when the SO does something. It's as if it's all YOUR job and then a generous or magnanimous husband or SO gets to feel like a big hero for doing something of his choosing during a convenient time frame, and expecting gratitude.
What your SO is doing is called PARENTING. It's not babysitting or "watching" the baby - that's what grandparents or hired sitters do. If your SO has a 6 year old child, then he should have six years of experience with this.
I think it's really important to appreciate what the other one does. Possibly he feels you don't appreciate the time & effort it takes to plan and prepare meals plus clean up. Obviously you feel he doesn't appreciate what it takes to tidy up and do laundry. It also sounds like you don't see each other that much, given your work schedules - so maybe there's some feeling that "I'm in this alone" on both your parts. Is there some old baggage from his ex and his feelings that there is stuff she didn't do? Maybe he feels like the financial provider there and it's up to her to do everything in the house? I'm trying to give him the benefit of the doubt because my first instinct was to criticize him.
I would suggest that putting in a load of laundry is EASY. Same with putting it in the dryer. It's the folding and putting away that takes the time. But it's a good "in front of TV" job after the baby goes to bed. My husband and I still do this together, and it just stops the complaining. It makes it easier to fold sheets and sort socks if there's a second person too. We also unload the diswasher together - he takes all the stuff that goes in the cupboards on the left of the dishwasher, and I do the right. If it would make a huge difference in your exhaustion and make you more available to him so you could have some couple time, that might be a good incentive.
The best thing to do, I think, is to switch jobs for a couple of days. It makes each one more aware of what it takes - things that look easy may not be, and things that look hard maybe can be streamlined. You may find that there are jobs you absolutely HATE, and he can take those on - and vice versa. Then you take the rest of the jobs and split them - either divide them up, or alternate. Also figure out what you can eliminate entirely, or simplify. Maybe you just don't make the bed every day. Maybe you get some baskets or bins in the family room or wherever the toys tend to accumulate, and stuff just gets tossed in instead of put away in a more organized way. Now that your child will be more mobile, the mess gets worse! We had a bin for "vehicles" (toy trucks) and one for "building things" (blocks & duplos) and didn't worry about sorting by each type of toy. Our son actually became more creative by combining toys in new ways - mixing train tracks with Duplo structures and cars, for example.
Now that you are working part time, he may feel you have to do more, and maybe that's okay. But also, just BEING with the baby is a job and very important. It's really important NOT to completely measure people what what evidence they can accumulate of having accomplished tasks.
If this doesn't work, then I would suggest you go away for a day or two - visit a friend or a parent, and leave him to do everything! He also has a child, and I don't know if he sees her separately from your home life (if so, that "counts" as work he is doing) or if she comes to your house and you are doing everything.
Good luck sorting this out!