So from a pure legal standpoint. She has a much better case against the school then the cops.
Assume everything she said is either true, or close enough to true. Then she can post whatever she wants. But Civil court is about making someone whole. The cop came to her house and told her to take it down or he would arrest her. She told him to buzz off, he didn’t arrest her, the only people who knew that happened were in her house. She suffered NO damage by the cop doing this so there is nothing for the courts to do in this case.
Now -The cop told the school (And depending on what he told the school, that was prob legal as well. For instance -If the cop said “You have a student saying she recovered by Covid-19, despite never having a test that said she was positive” - well, thats a true story so whatever).
The school, on the other hand, slandered her! They sent a communication out that basically identified her (By repeating details on how the student said she got it) and claiming that there was “NO TRUTH” to that. That was false - They didn’t know there was no Truth to what she said.
If I was on her side -I would argue that they knowingly, or should have known, that they couldn’t say there was NO truth, and they sent out that communication to all in a effort to dismiss her and create a reputation of her being a liar, to her school mates. That would be the only case I see where she can show damages and show she must be made whole.
On the flip side -If i worked for the school I would argue that i did not mention any name or give any personal detail, and that my communication was based on what I was told L&E at the time.
Meanwhile -If I’m the Judge - I tell the cop he was wrong, but let his employer handle it since no damage was caused to her by him going to the house, and I tell the school system to send out a email to all saying "Correction. We stated there was no truth when in fact her symptoms matched and while she had a negative test, she still was treated as a presumed case and if you had contact here are the CDC recommendations.
Just my 2 cents. Not a lawyer - but did stay at holiday inn Express back when traveling was a thing.