Doubt Amu will agree to abandon Kana, even if Kana tries to mind crush her.
If her parents don't agree that the best way to keep an eye on her is to have her boarding with them, Amu will probably at least insist on visiting her every other day. With the fog gone, the Scavengers' house is a perfectly habitable residence - and it is possible that Naomi and co. aren't permanently gone, just got tied up for longer than they thought they'd would be.
In fact, it depends a lot on what we get told happened to other Scavengers.
If the answer is "no idea", Kana herself will probably want to stay at the house just in case Naomi comes back or manages to send a message back to her. Since if the other Scavengers returned to find Kana gone, chances are Naomi would assume she got kidnapped or, even worse, betrayed them.
Since the upcoming event options indicate Kana would need some heavy persuasion to even have a medical checkup at JP's, trying to get her to submit herself to their custody does not seem like a practical option. Naoto might offer to safeguard her, now she knows that Kana is on the run, but the chances of Kana taking up that offer from a stranger she doesn't know and can't read the mind of is likely about the same as her wanting to go to JP's.
Alternatively, Utau might offer to house her. That may have better chances of being accepted by Kana than with JP's/Naoto, but Utau would need to convince her that the adults there at her place really have turned over a new leaf and aren't like how they used to be anymore.
I give it 50-50 odds on that.