THE END garment grid you must oversoul all fiends to obtain. You can check in Shinra's list to see which one's are oversouled and what needs to be.
I've got it, and quite frankly it wasn't really worth all of the work it takes to get that thing.
But for those who are working on Bevelle, you may want to know this.
To obtain The End G.G. you must OVERSOUL all fiends that CAN be oversouled.
You Do Not have to fight it oversouled, only Encounter it oversouled.
You can Oversoul All types of a particular fiend, by fighting one of them 13 times and then escape/flee,
Example: Fight 13 blue elements in Macalania, when the next one Oversouls Do Not kill it. As long as you do not kill the oversouled element, the other elements you run into will be oversouled.
(This is how to encounter the bosses oversouled without having to fight them.)
In case of the Red Dragon, you'll run into him on the 80th floor and up. For the Mega-Tonberry you'll eventually have to run into oversouled, try to keep a count of how many tonberries you've encountered, be sure you've saved recently and try to encounter the Mega-Tonberry when he's oversouled, It's not really easy to keep count, so you'll pretty much have to depend on luck.
The End GG is only something to work on if you intend on going all the way down in Bevelle. But as I've already said, I didn't think it was worth all the trouble, your oversoul's stay with you in New Game Plus, so you can still work on it in the next game if you want.