Unless things have changed in the last five years, as soon as you rec your attach-posting msge to undergo trg in Gagetown, you can staff a memo requesting a PMQ there. The Qs are in the town of Oromocto but are administered by the CF. You can rent a PMQ whether you're undergoing training or not. The army doesn't care where you live until you're on course and can be ordered into shacks.
You need to verify if that's 181 days of consecutive training, i.e. a single course that lasts a year. CAP + Ph III + Ph IV may not (probably doesn't) count as such. And the MFRC is not run by the military, so take what they say with a grain of well-intended salt anyway.
As for your dates, your chain of command should know by now when your CAP will start. It could be a few months from the end of BOTC, or it could start the Monday after you finish in St-Jean.