This response, if accurate, is so classic BA that I can't help but laugh. Sorry it was at your expense.

You should have more success elsewhere in the system. In my experience with CX, QF, and others, they see the status at check-in and issue lounge access cards at that time that eliminate the discussion of cards or anything else when you visit the lounge. You may have similarily frustrating experiences elsewhere but LHR is generally considered to be the apex of evil in the lounge access world. I can't remember having being asked for the actual card elsewhere, hopefully you will have the same experience.