In my experience, Batgirl is more so a test of what you know vs what you can deal with. Obviously, backdashing the cartwheel mixup on block is a must, as this will probably be her primary mid ranged tool. Once you can do that, then she'll probably start coming in with her dash > dash shoryu, which now it's time to start stopping her dash with down 2. At this point, now the game will probably revolve around her going for bolas and jumping trying to bait some shots. Bolas are probably best dealt with jumping, although she can delay the bola for a bit, but your jump is so quick that I don't think the delay can actually hit you on the way down.
Another fun thing is that if you delay a standing 3 after your strings and attacks that are severely positive (Foward 3 and foward 2,3 usually), she actually will be hit if she backdashes or jumps, so you can try and condition her to sit still for your 50/50's.
Lastly, another thing to press her on is commiting to doing her extremely unsafe wakeups, Shoryuken and Teleport. You can actually stuff the cartwheel with a well timed meaty, and there are certain setups that end in a crossup forward 3 that will avoid the shoryu and teleport while stuffing cartwheel. Of course this again comes down to delaying wakeups and other mindgames, but scaring her on a knockdown into risking it all is better than leting her run cartwheel on you for days.
I think that Batgirl is very good, but with your very good anti air and low shots, you really force her to take risks in the neutral game to try and score damage. Cartwheel is a really good midrange option, but luckily the mixup can be backdashed, which leaves you with an option against it. To me this sounds like an even matchup.
