First, thank you for helping Eddie back to health—we love to hear success stories.
Round pens can be used in many different ways. They can be good tools in training a horse to learn things like verbal cues, transitions and gaits. But, like any tool, they can be used appropriately or inappropriately. I am not fully familiar with techniques that use the pen for discipline.
Suzanne, my concern is that no matter how Eddie is being encouraged to move around the pen—whether this is via positive reinforcement, negative reinforcement, or punishment—having a horse bullet around a small round pen is a recipe for injury. The strain on the inside legs can be significant, and a stumble can become much more than a stumble as there is little or no straight away to recover.