I think i'm braindead from the amount of time i've stared at this. What is the equilibria in each of the scenarios in this? and preferably the subgame perfect equilibrium? Is there even any, as player 1's choice to bribe can pose different payoffs at the end?
When looking for equilibria, I would start from the end instead of the beginning. What are player 3's optimal choices, assuming player 1 bribes and player 2 accepts? For each of player 3's choices, what are player 2's optimal choice(s)? Working backwards lets you decide what player 1 will do in each situation. (In theory, you're asking "Given any possible sequence of player 1's and player 2's choices, what are player 3's optimal choices?", but in this case there's only one branch of player 1 and 2's choices where 3 gets a choice at all).
Edit: a Nash equilibrium needs to be a complete contingent plan. You need to say "P1 bribes, then P2 declines, then P3 doesn't get a choice but if they did they would punish P2" (you could write this as $(B,D,P2)$). If P3 plans to punish P2, there's only one optimal strategy for P2, whereas given P3 plans to punish P2, then there are two optimal strategies for P2. A Nash equilibrium only considers what happens if one of the players changes their move, not if (e.g.) P2 changes their move, then P3 changes their response.