I don't ride barrels, never have, so take my opinion in that context. I think that when a horse has difficulty turning one direction it isn't because they don't want to turn that way. It's always a physical reason. There is some pain or stiffness that makes it hard to go that way, so it's not them being stubborn or bratty, as some would say.
Therefore, I would go away from the barrels and work on getting him supple and comfortable turning both directions without the pressure of the barrel race.
As for the bit, I have used a tom thumb, just like the one you posted, and I really didnt' like it at all. I found that it was hard to do any sort of direct reining in it and it encouraged my horse to come behind the bit and then just plow right through my hands. So, instead of having more control, I had less.
What sort of bit is most commonly used for barrel racing? Do most barrel racers direct rein or neck rein? I am not sure what bit to advise you on,. But I think working on plain and simple suppling him to the right OFF the barrels would be a good idea.