Is it possible for you to meet right across the border in ogdensburg, NY and get married there? its $40 dollars (plus a fee for the officiant)
http://www.health.state.ny.us/vital_records/married.htm
you would both need a birth certificate and a passport and you have to wait 24 hours to get married, but its a lot faster and cheaper than doing it in Canada. IF you go to the border with the marriage certificate in hand, with proof of funds to support yourself for several months, and proof of health insurance for the same amount of time, they will probably give you a visitor's record that allows you to stay at least until you can file your paperwork. When they have proof that you've filed they'll give you a longer record.
I got reasonable traveler's insurance from https://www.kanetix.ca/travel-visitors-to-canada
The worst thing that will happen is that they will deny you and you follow the plan you mentioned above with the marriage bit already taken care of!