They'll be using somewhere like Bitpay most likely which sets the price the moment you're charged.
Correct. As you can
see from the government website, they are using
Coinberry, a Canadian company which operates both a crypto exchange and a crypto payment processor, offering "instantaneous no-fee withdrawals of Canadian dollars".
If you read the
news stories surrounding this, they are doing it simply to get away from the high fees charged by credit card companies. More adoption is always good, but I can't see a huge number of people lining up to give the government details of which bitcoin addresses belong to them.