I know this thread is old, but I'm facing this same issue. I have an Airbnb for a few months so I cannot provide a lease agreement. But I notice that one of the alternatives is a Canadian driving licence, and to get a licence you only need to show a bank statement. I think (although I don't know for sure) that you can use a mailbox, which gives you a residential address, to receive a bank statement which then can get you a licence, which you can use to apply for healthcare.
Can anyone comment on the ability to use a mailbox address to receive correspondence sufficient to obtain a Canadian driving licence?