Yes, it is possible to train with a harness--any size dog, actually. Don't let anyone pressure you into using a collar. Just because this trainer uses a collar, doesn't make it right for every dog. Some dogs do fine, some dogs get injured trachea, and definitely a breed with issues such as Cavaliers should never use a collar. You stick to your guns, and use the internet to reseach training techniques. Use the class as a socialization class for your dog, since this is the most important part of it, to prevent fears of strange dogs and strange people. Positive reinforcement with praise and/or treats goes much farther than any collar can.
We don't see things as they are. We see things as we are. Anais Nin