Yes, you can mime your own limit break.
Since I'm talking out of my ass, I'll go action by action for you:
Battle Starts
Cloud uses the W-Item command, to use two Hero Drinks (increases Power / Speed / Defense / Magic Power / Spirit / Luck) on Barret.
Barret uses the Ungarmax limit.
Emerald Weapon uses Emerald Shoot on Barret.
Game switches to show at least two mastered Mimic Materia equipped to Barret, connected to at least two mastered Command Counter Materia.
Barret uses Mimic, to do Ungarmax again multiple times via the Mastered Mimic + Command Counter Materia combos he has equipped.
I beat Emerald the first time with Lucky 7777, then a couple KotR. Nowhere near as quick as this, but it was right after the game came out.