It's a heck of a lot easier to clean a small place than a big place with another adult in it constantly messing it up. Just saying.
Would you want 50/50 for yourself with DC if you split? It would make your life easier if you only had EOW, although I realise you may want to keep them close as much as you can.
If you're that incapacitated you can't walk around your own home, you should claim PIP. As well as hassling NHS to do something, there's procedures they can do that helps improve the pain for some people for a time, also various painkillers that may work at least a bit, you have to let your GP know how bad your situation is otherwise they'll write "not applicable" under the how does it affect you section of the form DWP wants them to fill in.
You sound like you need a care assessment from adult social services too. They're obliged to do things like help you leave your partner or help you care for DC. You may be expected to contribute financially towards whatever help is provided but it does sound like you'd qualify for some support hours.
How much stress is this relationship causing you? Because that won't be helping with anything.
If you want to split, then split, and if you can't cope alone then deal with that through official channels, instead of getting back together with someone who doesn't really care about you. If that's why you're together (and you make it sound like that's the primary reason), he's only doing it to avoid child support or solo parenting young DC, so chances are he'll leave you for someone else eventually anyway.
You can claim UC too, if you can't work.
I don't agree with those saying he needs a break, it's hard on him, etc. That may be true, but this is a new disability that you haven't adjusted to yet, mentally or practically. It would be different if this had been going on for years, but it's 4 months and at this point you need his support if you're together. It's your first Christmas like this, it's a new situation that you're still getting your head around and no practical support has been put in place, including adequate pain relief. It's not the time for him to go off on his jollies and leave you alone with 3 DC under 10.