In RHATO #9, Red Hood and Batgirl have a small crossover, no big deal. But Jason knows she is Barbara Gordon and they seemed to have had any sort of interaction before.
In Batgirl #0 (issue that I loved, btw), we got to know that she was only Batgirl for a year... so she probably quit before Jason becomes Robin.
If my assumption is right, it means she remained close to the Bat Family during the years she was paralyzed (and the year when she stopped being a vigilant). TOO close, to all its members. But I don't believe it was just due to her older friendship with Dick and Bruce. Maybe there's something more in it, a part of the story that we still don't know.
With a BOP issue dedicated to explain their bond with the TT and Batgirl showing up in TT #15, we may get more pieces of that puzzle.
P.S: Just my thoughts, no need to stone me or anything ¬¬