It does not sound to me as if your RT needs much other then more hunting, and less time in proximity to Chickens.
RT's capacity for learning the easy way, can make them moderately effective on Pheasent.
A second year bird I had would just keep following it (losing ground all the way) until it put in.
She would then stay near, which kept it from running.The dog and I would arrive and jump it again.
A couple of reflushes and a rooster may decide to fight it out on the ground. That never turned out to be a wise decision.
The Hens were much harder to reflush preferring to lay (very)low.