In a transparency game (the default), psionics is affected by Spell Resistance, Dispel Magic, etc, and so is for all practical purposes just another form of "Magic". Binders have all supernatural abilities, though, so I don't think they'd count. We could also come up with some arbitrary threshold, like the challenger can't have higher than 6th level spells/powers, to allow things like Arcane Archers, Psychic Warriors, and heavily multiclassed gishes to give it a shot. Otherwise, yeah, I don't know if it would even be possible.
Also, using scrolls and things via UMD could be allowed, which would mean that the challenger could basically be a pseudo-caster, but this would also have the advantage of being a counter argument to the stupid "yeah, but it's just the spells that are overpowered" argument(s).