I'm guessing you do not want her there when you are not there OP, which suggests you do not trust her. Which is fine.
When I flat shared one flat member moved her dp in without consulting us. He was there a lot, often without her as he worked off shore half the time.
It was very annoying, there when she was not there. I'm guessing this is the main issue? If she is there when your son is there she is his guest, you don't want her dossing at your house, and I don't blame you.
Just tell him your rules, your house.
He should not be arguing with you like brother and sister, that shows no respect for you.
He should pay rent regularly, he will never learn to pay rent for real if he only has to pay some of the time.
Lastly, if he works very hard but does not have enough money to pay rent and contribute for food it suggests he is not earning enough. Can you encourage him to get a better paid job, to under-go training etc while staying with you with a view to getting a better salary and moving out?
What her parents think of him is not relevant unless she shares this view or you secretly think he could do better work-wise etc.
Good luck. 