Firstly, sorry if this seems harsh, you must be having a really hard time right now
and I hope things get better soon.
Of course YANBU to want/need help but a few things jump out of your OP.
He's already been off for 4 weeks and done little more than some light babysitting!
Baby is bottle fed yet you have done 100% of feeds/wake ups during this time - why wasn't he taking a turn?
It was paternity/parental leave not a holiday. The problem is the time has been and gone to raise this. Now he's back at work doing 13 hour days it's understandable that you would be doing the brunt of childcare and sleepless nights. But you should have just had an easier month with full time help to prepare!
The mother thing is just a distraction here, although clearly the straw that broke the camels back for you. You are completely right about it, he should be prioritising time at home, but because you have 'gone along' with the way things were so far I don't think he sees this. Perhaps he thinks you're coping just fine.
Impossible to know from this if he's just thoughtless or there's more to it than that. It's hard to think rationally when you are so tired. Try to have a sit down together and discuss what you both need? Spell out calmly to him why the mother thing upset you so much. And perhaps tell him other fathers do a LOT more on paternity leave (if you can do that without an argument).