Thanks; I thought I remembered Tim enjoying pizza and secrets just before Ted arrived. (I just found Dixon's script for this issue at Comic Book Script Archive.)
I recall moments in Batman: Cataclysm (1998) when Barbara doesn't know "whoever Robin is now," but knows Dick knows—and clearly dislikes it. This Birds of Prey issue is July 2000, and she's still in the dark, though comfortable with the kid.
I'm glad to know there wasn't a story behind Barbara's knowledge of Tim's name or a dramatic reveal. I've been looking for that moment, and wondered if I'd missed something big.
Speaking as someone who shares Barbara's (and Tim's) pleasure in commanding information, I wouldn't be surprised if, when she does reveal that she knows, she leaves the impression that she'd known longer than she actually did.