Yeah checking for dust in high places is beyond stupid of him, he does sound like he's being a dick. Getting his mum to weigh in was cowardly and childish.
However I've read on here the principle of everyone getting equal leisure time. It does seem unbalanced at the moment. Baby classes and coffee afterwards is leisure time for you (in my strong opinion, babies below about 9m do not get any benefit from them apart from the indirect benefit of a happy mum).
So you're getting 2h+ of leisure time a day. I think it's therefore OK for your partner to want similar leisure time in the evening, even though he is being a dick about it.
Prioritise a quick cleanup of the main areas, it should only take 15-30min a day and you have many hours at home.
Do try the singing and talking thing if you don't already. I was able to pop out of the room for a few seconds with my dc in the bouncer in the next room, by singing and peekabooing frequently.
Cleaning the toilet for example: quick spray of the cleaning stuff then pop your head back to baby, sing and chat, then quick scrub and wipe, and then back to baby. Your baby will literally find this fun, I remember mine laughing when I did this and they were both velcro-like breastfeeding monsters. But they thought I was doing it to entertain them rather than getting housework done.