Could anybody can spell it out for me what does the sentence " Don't pull that card on me" mean? Thanks.
This is a guess: it means 'don't try that trick on me' or, 'don't try to gain the upper-hand in the conversation like that'.
But, since she is entitled to see a copy of the tape anyway, it's likely that Tim is using the expression needlessly. He is not so intelligent, after all.
Hope this helps.