A Fell Beast could be a tricky target for a large ballista, but as Legolas proved over the Anduin, they aren't invincible. But of course I doubt that Fell Beast was as wary for flying arrows as one would be in a large scale battle and Legolas's hit may well have been a fluke. To insure a good hit you must attract the targets to something they would go for anyway... like a King of Rohan. Make sure every top general or leader has a poisoned dart he could shoot the Fell Beast with... and a fast horse to get away. Of course it's not as if in doing this we're endangering the most important men on the battlefield because the Nazgul would go for them anyway. Why not equip them to protect themselves and countless others.
Now as for discovering what ails the Fell Beasts? Ummm. Ask them? I'll get back to you on that one.
