> Do you just let Umbridge into your house and hope that the quick look around is enough to close the case? Or do you recognize you're dealing with an Umbridge and not take the chance?

In the US at least, you absolutely do not give consent for anyone from the government to come into your house. It can only hurt you. Call the police and tell them someone is trespassing. Call your lawyer and explain the situation.

Children should know it is inappropriate for adults they don't know to ask for details about their home life. Any adult acting in good faith will always make it easier for a child to contact their parents without negotiation. "Yes you can use my phone" is the only acceptable response. No "come over here first", no "let's talk about where your parents are".