I've gotten Squall to use Lion Heart by both getting a Perfect Renzokuken, and also by missing some. His "crisis level" can't have anything to do with it either. Since when I fought Ulti's final form Squall was at 1 HP AND in Aura status, and he still never used LionHeart ONCE during the entire battle with her final form.

The bottom line is that it is totally random, and nothing you can do will affect it significantly.