Scizor will be good against your rivals, because poison does nothing on it and it resists psychic. It won't do terrible in the dojo, because fighting is neutral. I'm not good at movesets, but I'm gonna throw one out there.
Bullet Punch, (Like you wanted)
Swords Dance (Sweeping capabilities)
X Scissor (Stab)
False Swipe (Catching)
The only weakness is fire, for which the only coverage is sand tomb (which kinda sucks). So I put in false swipe.