Aperture, aperture, aperture.
If you're not going to blind them, and you need the fast shutter speed, increase your aperture as low as your camera can go (looks to be f/2.7). Combine that with bumping up the dreaded ISO speed to something you can deal with in NoiseNinja, and you should be able to hopefully get shots that are salvageable.
In addition, you can always try to lighten the shots up in post-processing if they are only kind of dark. Sometimes you're able to work wonders in post on a dark shot.
You're pretty spot on about needing that fast of a shutter speed for freezing the motion, and it might be possible to capture movement in the arm with lower shutter speeds, but it depends how much you want the action frozen. Not sure how much that helps or if I just rambled on a bit here