Benchmark was removed for Batman AA in Steam when they had to release a new version since Microsoft dropped support. I think I benchmarked 165FPS avg for my GTX680 (GTX770) rig with everything maxed except PhysX off. There's some numbers that might help you on this list (again, do NOT use PhysX):
http://www.gamephys.com/tag/batman-arkham-asylum/
On a desktop with a GTX285 they got 155FPS with PhysX off at 1280x1024 (4xMSAA) so you should again have a good experience. Hard to find benchmarks for your graphics.
*You should aim to get 60FPS if possible with VSYNC, or if screen tearing isn't too bad at least 40FPS with VSYNC OFF. You do not want to get too low of a frame rate as it's more important for the fighting to feel smooth than look better.
**RadeonPro has a feature called "Dynamic" VSYNC that works the same as Adaptive VSync for NVidia. I suggest using that, then tweak the graphics quality so you mostly get 60FPS (if possible).
Other:
If you support 1920x1080 but can't maintain 60FPS then 1600x900 might be a better choice. You'll have to experiment.