Going into this thread, I was originally assigning the win to Batgirl, but reading over the scans gave me something to think about.
When it came down to it, I felt Bullseye would take a win if he began with ranged shots, and would lose or hold his own in hand to hand combat.
The feats of Batgirl dodging failed to impress me greatly. It felt like touting that Wolverine beat "x amount of hand ninjas".
Meh.
However, seeing Daredevil knocked out with a paper airplane, and knowing of Bullseye's throwing feats that he performs casually gave me cause to think that he could hit Batgirl in a ranged fight. And honestly, he only needs to hit her once for him to then take an edge.
Anyways, I vote Bullseye. Phil failed to convince me that Batgirl outclassed Bullseye's history of feats.