Charged Aerial Shooting > Satellite Aim > Bullet Squall seems to be the best combo so far, and it's definitely the coolest-looking string too
For Chain Finish, I usually keep an identical weapon in the 2nd palette with 3x Satellite Aim - the damage is amazing with it.
I've never really been short of pp, so I don't see any need for PP Restorate...
The common scenario when playing Gunner is that you'll have plenty of PP but struggle to find an opportune time to close in and dump it all. Especially if you're trying to fill up your Gear; you'll want to be as trigger happy as you can to fill it up.
On to Chain Trigger.
The default cooldown time is horribly long in reality, so i'd recommend maxing it out over Chain Finish if you're only going to do one of the two.
I have to point out the biggest flaw in the skill, though...
Chaining with Mechguns is pretty horrible on any moving target because you'll eventually find yourself in the following situation:
(i) You've started the chain but the enemy is too far away to get a PA off, so you'll have to run in for it. But wait! if you stop attacking and start closing the gap then chain will break!
So you're only option is to step forward > attack > step forward > attack > step forward > get punched in the face, fall over and lose the chain. And i'm sure it's happened to all of you
Gunner really needs the ability to jump forwards.
What i'd recommend is building the chain with a rifle (yep, you can do this), so that you can still shoot whilst strafing the target and positioning yourself accordingly. You can then switch to a Mechgun for the PA, or you can use something like Sneak Shot instead, if you want.