That's a good question. In the past I was a Costco cardholder but never BJ's, because I have never lived in an area which had their stores.
If the Barclays BJs credit card operates like the one from Costco, than your picture and membership ID are printed on the back. In the example scenario you gave, I would think you would just be able to flip over your BJs Visa card to show your membership and then pay using a different credit card if you wanted to. That said, I Google Imaged "BJs credit card" and could find the picture of its back, so I am unsure if the membership info and photo is printed on the back.Something to point out about the rewards
though is that unlike Costco and Sam's Club, you can use any credit card at BJs - Visa, MasterCard, American Express or Discover. The BJs Visa gives 2% at their stores and 1% elsewhere. If by remote chance BJs is not classified as a "warehouse store" category, then you would be better off with a credit card that gives 2% everywhere like the Fidelity card offers
Disclosure: I am a moderator/paid staff of this site, which does have advertising relationships with some credit cards that are discussed and linked to. Regardless, anything I say is my honest opinion.
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