One rein stops are meant to be done in a snaffle bit, not a curb. A kimberwick is a a curb bit. Curb bits are designed to increase collection, collection is the opposite of disengagement which is what you are trying to achieve with a one rein stop. Any sort of leverage bit will be less effective at teaching a one rein stop.
I have to agree with Oh vair oh, if your horse is bracing against the bit hard enough to cause cuts to his mouth, there is a big hole in his training. It might be worthwhile for you to spend time going back to a few basics rather than just changing bits.
This horse does need more training. He has made a lot of progress in controlled/easy environments like the indoor ring. When taken out in more stimulating environments such as outside doing gymkhana games or on trail rides with large groups of horses, he obviously does not respond as well yet.