The more you post the more red flags are waving all over the place @yan79!
Sorry, I know this is not what you want to hear, but...
He wants you to contribute 50% to the bills which you've said is impossible.
He wants you to work full time, but again you've said this is impossible due to childcare.
You stand to lose over a grand in benefits by moving in with him. Has he offered to contribute to cover any of this shortfall? Nope, thought not.
You are giving up your independence and putting yourself and your children at risk of being homeless if you split up.
If you try and talk about things you say he's short-tempered, spits out his dummy and storms off for a couple of days.
Do you REALLY want to move in with him? What if you move in, have a row and he chucks you and DC out? That is a very strong possibility.
He doesn't sound kind, or considerate or with any of your best interests in mind.
You need to put your big girl pants on and have a VERY honest discussion about how and if this moves forward. You CANNOT afford to to move in with him on his terms.
Good luck, but I'm guessing he'll lose his rag and storm off. He CAN afford to do this on his own, he's already said that.