If this woman's intentions were honourable, or even if she was just there to act as a visitor on the son's behalf, then she should be open and up front about what she is doing.
To sneak behind your and the care home's 'backs', claiming she is a relative, just doesn't sound right.
If she has nothing to hide then she wouldn't mind another person being in the room on her visits, or being honest about what she is doing and why. By now she should be clear that your father really is unwell, and is in the care home for genuine reasons, whether he wants to be there or not, and that he is being cared for properly.
What else could she report back to the overseas son?
If she is there for genuine reasons I have no reason to stop her visits but in the presence of myself or a member of staff so yes i'm with you on this why lie in the first place. There is nothing really to report back to him about unless there is another reason behind her visits and that is what is bothering me. Also he has a phone why doesn't he phone the home he has before and they'll tell him everything he wants to know its just silly.