It's not obvious at first, but this is where the discrepancy button comes in handy for unique situations like this. You highlight the name of the card the stripper gave you, as well as the same name of the man's passport. If not, it's somewhere along those lines; I know for a fact that one of the correct highlights is the card.
Most of the time, story writing in games is an afterthought until the mechanics and tech have been laid down. That's just how it is.
I feel that Tomb Raider was made with two different visions: one focused on the game and the other focused on the story. The conflict between them is clear with the whole "first kill" ordeal, but I don't think the story was a complete failure.