I'd like to explain a bit how Battle on the Big Bridges second phase works and how you can phase skip the final part (where he frogs the whole group, then shrinks the whole group). There have been a few PF groups that know how this works, that use a solo healer, and down Gilgamesh in probably less than 5 minutes.
The key here is his move "Rocket Punch". Rocket Punch randomly selects two players to be hit with the confusion debuff. During this time, they run around and use auto attacks on other players. Usually a Black Mage, Summoner, or Bard is chosen (ranged dps). The debuff can be removed by healing the player. You want to be sure that you heal the debuff off of any physical damage dealing player ASAP (Bard, Monk, Dragoon, etc.) as a hard hitting Bard can potentially kill a player very quickly. Summoners and Black mages won't hit a player for more than 300 damage per slow auto attack, so they pose no threat being confused. Just make sure you kite the player out of damage.
"But isn't it a dps loss to leave one player confused for the second half?"
No. Avoid using group heals, avoid the whirlwinds, be mindful of his dives, and healers - don't just stand there! Throw in a few offensive spells! This fight is NOT hard.
The confusion debuff lasts for nine minutes. So long as its not healed off the player, Gilgamesh will not use frog and minimize on the entire group. It is more of a dps loss to have the whole party immobilized the last 10% (or less) of the fight than it is to have one caster not casting.
"What if there is no caster in the group?"
I've not encountered this specific instance, ever. I'm sure its possible. My suggestion would be to be mindful of which player is confused, and to heal the person getting hit through the auto attack damage.
If anyone would like to comment or add anything to this, please do. I just hope DF groups don't continue to laugh at me when I suggest something "crazy" like this!