Many people use this strategy; however, in order for it to work the annual fee should be waived the first year.
Unfortunately, unless you have an offer I'm not aware of, I do not believe that the annual fee is waived for the BCP right now.
When I received my BCP in July, the annual fee posted within the next month, so there would have been no opportunity for me to keep it a year and then downgrade to the BCE.
There are ways to make the BCP pay for the AF. Even if you don't spend enough on groceries, throw in a $500 gift card at the grocery store once a month and it will pay for itself plus more.