Credit goes to lethalsquirl for making a FAT that actually works—I just improved it.
Change list:
* Tipped the slide beam back as per 
here (Item 1).
* Raised the counterweight up to the level of the release mechanism.
* Lengthened the arm and increased the arm ratio
* Lengthened the "sling" and made it one piece—a chain would utterly fall apart under the tremendous throwing force.
* Made the release mechanism larger so that the sling will not simply pass through it. Again, compensating for Phun mishandling large forces.
* Made the load-bearing parts (wheel and slide bar) denser, again for the same reason.
* Made the counterweight several pieces for rules reasons. Also made it a lot bigger and heavier.
Result: Throws now hit the killer plane, ~1km away, 
near the peak of their trajectory.