I’m the sort of person who cannot relax amongst mess
So he has you over a barrel.
He can opt out, because you pick up the slack.
You can't nag him into being a tidy person who cleans stuff.
There are three options:
1 You accept living at his standards and learn how to turn a blind eye.
2 You clean to your own standards and learn how to deal with the resentment.
3 Or you split up.
Assuming you don't want to split up, there are a few strategies that might help:
One - don't monitor who does what jobs, monitor down-time. Make sure you get equal leisure time to go out, or time sitting on the sofa looking at screens.
If necessary, take it to extremes: when he is sitting on the sofa, you sit too. If he asks when is dinner, you say "shall we look in the fridge make something?", then don't get up off the sofa until after he does.
Two - don't be the primary parent. This means letting go of the reins. Let him do it his way, don't micromanage. Go out and leave him with the baby, without a list of instructions of what to do or not to do. Let him decide what to feed baby or if baby needs a bath. He will work it out.
Three - don't be the default parent. This means at weekends and in the evening (i.e. not during the working day) he has to check with you first before he can make plans to go elsewhere or be involved in something not-baby-friendly (like cleaning out the garage). He can't just go ahead without checking with you first that you are free to do the childcare.
Point out to him every single time he has gone somewhere or started a task and just assumed that you will be on childcare duty.
If he persists, just get up one morning and go out without any warning - leave a note on the kitchen table or send a text to say you have gone for a walk and will be back "later". Let him experience the feeling of suddenly being stuck with the baby with no notice and having to change his plans. If he doesn't get the message, repeat as needed.
Four - don't give him bits of jobs, give him the whole area of responsibility. So you don't say 'can you put a wash on' - either all laundry is his area, or it is yours. Or you don't say 'remember to buy loo roll' - either household supplies are his area or yours. You are not his manager to hand out tasks as and when.
This means a sit-down discussion where you agree who does what - laundry, bins, car stuff, food shopping, other shopping, cooking on certain nights, dishwasher, cleaning bathrooms, garden, etc.
Then most importantly, you don't tread on his territory. You leave his areas of responsibility to him, and let him fail, without stepping in. Grit your teeth, bite your tongue, you don't get involved if it is his responsibility.