kakashi copied that gate opening jutsu during lee vs garra match in part 1..so there is possibility that both of them will use gate of death , after all they are eternal rivals.
Kakashi knows how to use gates U_U He can use them at will but doesn't do it because it's a forbidden tech. Gai relies on the Gates because he doesn't have much options whilst Kakashi has the most powerful Mangekyo ever, 1000+ jutsus, 4 elements and other things U_U
Gai wouldn't throw his life away before using the forbidden weapons Madara spat out, and the idea of copying a kenjutsu is false. If i were you I would change where I get my information.
I think it'd be cool if he used the 2nd or 3rd. I doubt he can get 4+, he's nowhere near as dedicated to Tajiutsu as Gai and Lee. Would be a total troll to those two if he did it.