He's not friendly towards me. Never engaged in chat, there is always something he needs to be doing.
He comes home and cleans, he doesn't take any of the parenting pressure off me.
Mutters and makes comments about stuff under his breath.
OP you are where I was a few months ago. I was at total breaking point. You are describing my DP to a tee, including the fact that when outside the home or 'doing something' with the toddler (just playing around at home is not 'doing something' apparently), he's actually a wonderful Dad.
I don't know what to tell you to do. I still don't know if we'll make it longer term. But I fully understand and feel for you. I understand why at this point you don't give a shit if he has a MH problem or not, because he's giving you one with his closed-off, selfish, unreasonable behaviour. I understand how it feels, like you're just an inconvenience to be worked around in the perfect life he is always trying to build for himself. To feel so unloved and unappreciated and unengaged with.
What you said about the muttering under the breath. No matter how many times I tell DP how much him banging around in another room muttering and huffing made me anxious he just WOULDN'T. STOP. DOING IT.
Sorry, I don't want to make this about me, but I really really feel you. Don't listen to everyone getting up your arse about 'in sickness and in health'. You matter too, your feelings matter, and with the best will in the world you can't help someone who refuses to believe there is anything wrong with them and puts it all back on you. 