'All the responses here say no based on finances, loosing benefits.
'Do you love him? Do you see a future together? If the answer is yes, then find a way to make it work. You only live once and deserve happiness.'
Because on Planet Reality, we all have to make a living. NO, OP! Just say NO. He's a 50/50er and you cannot afford it. He is asking you to give up all your financial security to make things easier for him. You need to imagine the worst case scenario, because the fact is that you are a lone parent, and that makes you and your children vulnerable. So what will you do if it doesn't work out and he tells you to get out of his house, even after say, 5 years. You'll be looking at homelessness (best of luck finding a landlord to take partial benefits and children) and Universal Credit.
NONE of his solutions involves taking responsibility himself - get your family to do childcare, get your ex to pay more, get more money at work. It's about making things easier for him.
Your moving in has ZERO impact on his financially and major consequences for you and your kids. LOL @ your getting rid of your car and just having one with him. It sounds like he wants more control over you, and this is exactly how he'll do it.
'That doesn't work for me and my kids just now. So I can't move in.' Rinse and repeat. Because moving in with him is a non-starter for you. Your primary responsibility is to your kids, not a boyfriend, and moving in with him puts them at serious financial risk. NO, NO, NO.