The only thing I would give advice on is going back a little more to the basics. I could be misunderstanding where you are taking him, but I will share what I was taught when I was running on barrels. You can disregard if I'm off the mark :)
Naturally you want to start with the trot, get to where you anticipate to "check" the horse and stop. Pushing him out at a trot again and repeat until the horse naturally starts to slow.
Same goes for the canter. Come at the barrel as if you are prepared to turn it, stop him where you plan to "check" him and push him out at a canter around the barrel. Repeat with the other barrels. This will help you acquire crisper turns, less "shoulder out" as he's doing. He's bending in his head, but you need to push his ribcage with it so he doesn't throw his weight out and away from the barrel. You want the focus and pressure on his haunches, not energy spent on running away from the barrel in evasion.
I applaud you running in a snaffle. I ran for money with my gelding (and won) in a d-ring snaffle. Good barrels are taught slowly, just like anything with a horse :)
The Fluffy Equestrian
Rhett | Hang on "Fiona" | Rockstar
Reg. TWH Gelding | Irish Draught Horse Mare | SSH Gelding