Satyrs are strange, strange creatures.
My military ignores them, unless they stand on the same square. The two will then push each other away, the military will chase a bit, then return to their post, even if they were told to chase forever. Marksdwarves are nearly impossible to get out there, as they'll kill everything else first, wasting their bolts (raccoons, ambushing goblins, etc...) before they can get to the Satyr. It's hard enough with spears, but they'll at least leave a trail of slaughter and continue on their way.
Now, caged Satyrs do cause new Satyrs to spawn. Once you hit any number above 15 though, they stop spawning. I may have eradicated all local Unicorns, which would suck (and necessitate fort abandonment, as the purpose of the fort is Unicorn totems, Unicorn bolts and Unicorn leather backpacks for the mountainhome! Also leather armors), but I won't know until I've eradicated all local Satyrs, if that's at all possible. In any case, new Unicorn do not spawn either when I hit about 17 Satyrs.
I tried building an execution tower. The result was that as soon as they were let out of the cage, they'd punch the dwarf that let them out and try to flee. This *did* allow my military dwarves to fight them, the issue being that some of my craftsdwarves would get injured in the process (non-fatal usually, but it does slow down my fort).
I've got a drowning chamber, but that's for goblins. If I kill the Satyr, I want their bones and skulls! As last resort, however, I'll fill it up with Satyr cages and then drown the whole bunch.
I'm setting up an Arena where I will put my master hunter, open the grate (so no one can escape) and then unleash all the Satyrs. Hopefully, stuff will proceed to die, as the Satyrs are apparantly hostile once released from cages. This is very time consuming though, and with my fort growing larger, I may abandon it if I don't get new Unicorns soon.