Because of that part where I have to fight Persona flying around shooting beams and stuff and it seems like the more I get hit by its attacks the faster the Party Wipe time limit goes down or I really do have less than 30 seconds to beat him before that attack. I really want to beat this version before I start collecting X-Cubes for my Hunter, but for the moment it seems impossible to stop him from using the Party Wipe attack on me and I'm only surviving because of Iron Will.

I just want to know the mechanics of that particular fight, because it's driving me insane.

It's just a dps check, you have to do enough damage to prevent that from happening.

Another way to survive it is to charge your Photon Blast during that attack as you are invincible while doing so. However like Macmaxi said, it is a dps check. If you are having trouble meeting the DPS check then it may be a gear issue or perhaps you need to adjust your rotation.