No, you don't have to use the card to book the flight to get the free bag. As long as the credit card is associated with your frequent flier number, you are good to go.

I traveled with 3 other people this weekend on one reservation (I have the Club Card) and I didn't book the ticket with the card; however, I was able to get my 4 bag allowance with no problem.

Hope this helps!
Have a good one.