I'm supporting Raikaria here. Priority Bulk Up passing is way too much, and goes beyond the scope of this Pokémon. About Rapid Spin, instead, I'd lean towards allowing it. I understand it is a pretty good move, but while CAP certainly can defeat most common spikers 1 on 1 (barring probably Deoxys), he is fairly easily countered by the common spin blockers, especially Jellicent. On one side, this will prevent Tomohawk from becoming broken, on the other, though, this will help greatly our concept. Not only Rapid Spinning certainly shifts momentum, forcing the opponent to play a specific answer to us (in this case, switching in their spinblocker) is exactly what momentum is about - restricting the opponent's options.
So yeah, Baton Pass for Disallowed, Rapid Spin for Allowed.
About the other pending moves:
Block/Mean Look for allowed: These moves are rarely - if anything - used in the current metagame. I'd like to see if Tomohawk can put them to good use (especially with Prankster)
Gravity for disallowed: I disagree with people saying Gravity is pointless on Tomohawk. Focus Blast and Hurricane are good enough reasons for using Gravity, and a Gravity/Focus Blast/Hurricane/filler (maybe recovery) set is not something which goes with the original concept, at all.
Pain Split for allowed: A fairly useless move on Tomohawk, due to his high base HP, still might see some use thanks to Prankster. No reason to disallow it (although if I were to submit a movepool I wouldn't include it).
Psycho Shift for allowed: I disagree with Raikaria here. Aside from the fact I don't see any flavor issues here, Psycho Shift is far from useless (and I'd like to underline that being useless should be a reason for allowing moves, not disallowing them). Taking status and spread it on the opponent's team is a really good strategy, and is among the most basic momentum shifting moves. Plus, it CAN be statused regardless of priority Taunt/Sub, especially on switch-ins.
Rain Dance for allowed: As Raikaria said, why not (although I don't care as much about flavor). I echo tomtom5858 here, in saying Prankster Rain Dance is a good way to shift momentum in a Sand infested metagame. Plus, Tomohawk looks to be a good switch into Tyranitar and Hippowdon, so it'll have an easier time changing the weather, compared to the genies.
Refresh for allowed: Again, why not. I'd rather use Psycho Shift, but Refresh is far from broken, so let's allow it.
Swords Dance for disallowed: Priority SD passing is even more broken, potentially, than priority BU passing.
Also, I'd like to contest the placement of Stealth Rock. I don't like Stealth Rock on this Pokémon, for a simple reason. While SR can certainly earn momentum, it's way too easy to fit it into Tomohawk's movesets, since he gets a perfect dual STAB. Plus, he can Rapid Spin, and can come in to set up SR repeteadly if needed with his bulk and maybe recovery. So Stealth Rock should be Disallowed.