who knows the secret IDs of everyone in the Batclan because why not
Hal was the Spectre at one point, and therefore damn-near omnipotent. It makes sense that he found out the identity of one of the major then-members of the JLA. And even if Barbara didn't want HalSpectre knowing who she is, she can't stop the Spectre. Computer skills are worthless when your opponent can turn your computers into swiss cheese ,literally, with a single thought.