I was stuck at the beginning of THP8 (the Shag Dad's Balls mission) for months because of the R2 trigger, and when I researched it, it seemed to be a common problem on PS3s after the 2.41 update (if I'm not mistaken, it's this one). But I found a way to fix it; I didn't see anyone talking about the fix online, so I decided to share it here. Basically, I connected my DualShock 4 and set it as controller #1, then I entered the game and connected the DualShock 3, which automatically became controller #2, So I held down the PS button until that mini settings menu appeared, then I configured the DualShock 3 as controller #1 and the L2 and R2 buttons started working completely normally. If that doesn't work, you can also connect the DualShock 4 as controller #1 and the DualShock 3 as controller #2, then after entering the game, turn off both controllers and then turn on the DualShock 3, It also worked as an alternative when the first method didn't work.
PS: I used a Dualshock 4 as a support to help with the fix, but I imagine another Dualshock 3 would work as well. So for the fix you need 1 Dualshock 4 and 1 Dualshock 3, Or just 2 Dualshock 3s.