Use what ever you and your dog are more comfortable with. I have had great success training and correcting with the martingale collar on my BC, but if your pup doesnt like the collar, there isnt a need to force it on him, that could make training a negative experience for him, which you definately dont want. and especially since its a health issue, a harness seems like abetter fit. you can still have the same success in training with a harness and a collar. Why dont you try emailing your trainer and just let her know why you dont want to use the collar and tell her you werent trying to be stubborn or anything, and hopefully, she will understand.
"An animal's eyes have the power to speak a great language."
"Who can believe that there is no soul behind those luminous eyes!"