Alright. You can open a bank account with the Coop over the phone, just tell them you want a basic account with a debit card. No problems. There's no overdraft or anything, so credit history isn't an issue. That, at least, is a start.
How do you get the money to do the shop? Does he give you cash each month for expenses?
I have to agree that I don't understand why one partner would choose to withhold money from the other partner. DH & I have separate accounts, but I have his debit card more than he does.
All the household bills come out of his account, and mine is more the slush fund - for incidental bills and savings and such. I access both accounts and transfer funds back and forth as needed. DH & I make the decisions together, but I handle it all as I'm a SAHM so have time free in the afternoons when DS3 is napping to go over the budget when needed.
I would let him "drip sarcasm"... he said it that way to shame you or embarrass you into continuing. Don't let it shake you. He obviously knows how to do the books, so let him do it. He can "drip sarcasm" til the cows come home, but do NOT give in. His business, his responsibility.
Let him make his own lunches. I can't remember the last time I made DH's work lunch. He makes his own. I bake stuff and make sure we have things available for him to put in it, but he makes it himself. If you start by having him make his lunch, after about a week, then say "oh, while you're making yours, can you put together lunches for the children while I'm doing (something else)? If you get stuck as to what to put in it, just ask and I'll let you know!Thanks!" 
Same with baths. Once the 3yo's bath is done, say "oh, while you're there, can you help 5yo with their bath as well please? I'm tied up here doing (something else)! Thanks!" Breezy, matter of fact.... 
See what happens. Never hurts to try different approaches.
Yes, you've let it go a long time. It can be more difficult to break long habits... but doesn't mean it can't be done. His behaviour isn't your "fault" but obviously you've let it continue far too long. It most likely was a situation where you didn't mind so much when it was just one child, but as the other two came along and the work you needed to do at home increased due to growing size of family, neither of you thought much about how much of a change it is until recently.