So the problem is, whenever he slacks off you pick up the tab, whether that’s the work, the mental load, or the bill.
The problem is also you are “asking him to help” - which implies it’s all your responsibility in the first place.
I’d probably drop him in it a few times a week send an SMS - “I had a bit of toothache and there was a cancellation at the dentist so I’ll be going out at 4.30 and will go via supermarket on way back, so you will need to do dinner, bath and bed for the kids tonight”
then another time: “my mum has organised afternoon tea on Sunday for Aunty B’s birthday so I’ll be out 11am until earliest 6.30pm so you’ll need to have the kids for the day. If the weathers nice you need to change the beds and wash the towels too. If not then can you make sure you get the bathrooms cleaned while the kids are napping. You might also want to get up early on Sunday and batch cook the spaghetti bolognese otherwise there will be no dinner on Sunday for you, or Monday for anyone.”
its not right that’s you have to micromanage him but I would at this stage be ordering him around very calmly. leave him a tick list in the kitchen for the day ahead. Make sure his jobs are essential for HiM - so put him on cooking, laundry etc. He may skip one dinner but he won’t skip ever day. When you shop don’t buy the things he likes and needs unless he’s getting through his own list of chores. Let stock items run out - tea, beer, toilet rolls (you keep yourself a spare in your handbag!)
Never, ever, ever clear up after him - that means, dirty clothes, dirty kitchen.
And start leaving jobs incomplete - for example emptying the bins, emptying (or putting on) the dishwasher .
Your answer is always “I was doing x y and z which are more important alongside doing lol the childcare and my office work; why didn’t it occur to you to do it? It’s not my job to do everything.” Then walk away.