You can't make any payments or go past the MRV Fee Details/Payment page until appointments are available and as you say, it really is just luck of the draw. I checked regularly every hour but happened to take a nap once when interview dates had been posted. After that I checked every half hour while the consulate office hours were open and managed to get an interview date.
Their hours are 7:30 a.m. to 4:00 pm. If you're on Facebook I also recommend joining the group "Montreal K1 Visa Applicants" if you're not in it already and have notifications on. Just like Visa Journey everyone is very helpful and will post if they see interview dates have been added.
ChainMaiden613 got a reaction from CoralMermaid0009 in Montreal Consulate - Scheduling K1 Fiancé visa interview/appointment