I assume you are proactively searching for alternate employment? I hope you are using your work computer and work internet connection to conduct that search. It seems to me that your "bargaining position" could be much improved if you can manage to find something else to do and tender your "two weeks" notice prior to your deadline for Caccination.
You might also look into cashing in any PTO (vacation time, sick days &c) prior to the deadline. I have no idea how Canadian law deals with PTO post-employment, but here in the United States typically you lose any/all PTO when you no longer work for the company.
I recall one job where I actually had two full weeks of vacation time due at the time I was planning to move on, which I gleefully scheduled immediately prior to giving my "two weeks" notice.

I appreciate the thoughts and concern, Mr. Pate. The fact is that the more I think of it, the more I have decided that an exemption likely will be offered. For a few reasons. First, I have every intention of writing a 15 page explanation of my beliefs. If you think my posts here are long winded- you have not seen anything.
Also, the HR lady who will be reviewing them knows me. We are not best friends- she is across the country. But I have met her on a few trips, and we have had supper a few times. Still- I think that will count for something (though I admit it should not and wonder about all of those people in positions who never deal with head office. Why is it fair that they should have a "disadvantage" to me.)
I don't mean to blow my own horn- I am replaceable (somewhat.) But I have years of experience, and there is not a person I can think of who could step into my role internal or external to the company and make things run half as good. Again- I think that they will be looking for a reason to keep me- and not a reason to turf me. Once again, is this fair to people who are in a position like labourer who may not be given the same consideration? Probably not.
I have the option of forging a certificate. My immediate boss has made it clear that the company will not be keeping them on file. He needs to look at it (and he will not be looking very hard). But that seems to be a way out that my morals will not allow me to take. My aunt told me when I was around 15 that I have always done everything the hard way and that life would be easier if I just took the easy road. I have not learned after all of these years.
As for vacation time- they have to pay it out. I was going to take my three weeks owing to me and not come in again. And maybe I should. But they can pay me out if they decide to part ways.
The funny thing is that they still have not given us a form to fill out for an exemption. I have heard from head office that I should get a note from a pastor. I feel like I should get notes from 5 or 6 to point out what a sham that request is.
Maybe I'm just acting like an ostrich. Maybe I'm willfully blind and just "hoping" that things will work out. But I think that I have logical reasons to Carrie Anna on this path and not apply elsewhere.