Snow probably knew what Haymitch's necklace actually was
I just have a feeling that Snow probably could tell that the necklace was a flint striker when he asked to look at it in Plutarch's library. It's because he himself snuck in a weapon disguised as a token for Lucy Gray so he probably can spot the same trick. However he let it go because he learned about Lenore Dove through it, which was a better piece of information to destroy Haymitch with if he needs to. He probably decided to do it right then even.