I asked some friends and it seems like it's the AOR. However, it seems that for Quebec in particular having the CSQ, the AOR is not sufficient for applying for the BOWP as we need the R10 that is the acknowledgment of completeness of the dossier, which takes a few more weeks (for example, I need to give biometric first, but the first appointment available is in two weeks).
As I understand, the restoration of status is really not possible if you switch to visitor but only if you let your status decay without having a legal status. So I don't think that is a possible course for the BOWP.
Anyways, I have an appointment with the lawyer next monday, but things are cutting very close for me.