If I had to fix Kylo, the first thing I'd do is give him some damage. They kinda forgot to do this the first time around, but I think it's necessary. Considering, you know, he's an attacker.
His basic is ehhhhhh. It's not awful, but it's not really meant to be the focus. I'd probably just put an extra DOT on there any time he hits a Jedi or Resistance toon.
Lash out is boring. It's just straight-up boring. He does some pitiful damage then turns into a bad Fives for a turn. I would make this a much harder-hitting ability which has a chance to take turn meter away from enemies, daze the main target, and give his allies Speed up.
He's basically cleaving through your ranks like a goddamn madman, that's gonna make a lot of people pause for a moment and re-evaluate their life decisions that led them to that point. At the same time, it's gonna make his allies fight harder, because they don't want him to do that horrifying stuff to them.
Outrage is kinda cool, but a bit messed-over by Protection. So, I'd simplify it to deal the extra damage to any enemy that had hit Kylo between his turns. Also, I'd make it do more damage. Are you noticing the theme yet?
With Unstoppable, I'd simply make it so that he has a much higher chance to reset his cooldowns, but only if he's targeting the person who damaged him. He's basically tunnel-visioning on the people who have actually hit him, then showing them why that was an astoundingly awful idea. Plus, this works well with the outrage change.