Hi OP
I think you're giving your children too much power over your life. You shouldn't end relationships because they don't want you to have one, and your friend is correct (to a certain extent) that you are the adult, the one in charge, and sometimes we make decisions they don't like.
Now, having said that, I don't think you should have this friend to stay, for two reasons.
The first, and most important, being, that you DID ask them, and they said no, so if you ignore that, it will undermine your relationship with them. They need to feel they can trust you, but if you're going to ignore their decision, what was the point in asking them in the first place?
Also, as your eldest is already having behavioural difficulties, this will only exacerbate them.
The second reason, is that I can't see how staying with you for 3/4 days, will actually help your friend. Where will he go when they're at home and not in bed? What help can you give him that you can't give him online or by phone?
If he's not got enough money for food, he needs to send less to his family back home, or you could do an online shop and get it delivered to him. You could send him some money for a top up on his phone. How is he going to pay the fares to travel from his?
I understand you want to help, so do what you can from your end, but because you asked your children, and they said no, you can't have him stay.
Sorry it's not what you want to hear, but I hope you're able to help him another way.