I personally am not a fan of any trainer that doesn't adapt to the dog and owner that it trains. One solution DOES NOT fit all. I also believe that trainers who automatically don't believe in treats are wrong. I look at it as "paying" for services rendered. I don't work for free so why should my dog
. It helps to think of it as "why on earth should my dog choose to work with me when there is so much distraction and other stuff more interesting). By using treats or other rewards and playing fun training games with your dog you then become way more interesting then all the other stuff and the result is a wonderful working relationship with your dog. I use treats in training with my dog and have a happy dog that loves to work. The trick with using treats is to not use them as bribes but to use them as rewards. The use of other rewards such as tugging or balls etc is an excellent way to train. I also don't reward for every single thing I ask the dog to do. The rate of reinforcement for learning something new is high but after that the reinforcements drop to much more random schedule.
I'm also not all sweetness and light for training. I understand that dogs need rules and am not nieve enought to believe that we can be all positive all the time. I do not believe in using any "punishment" that frightens, stresses or even bullies. I am a big fan in using "NILF" training (do a web search for Nothing in Life is Free dog training).
If you need the halti then tell the instructor that is what you are going to use to complete the course. The halti can also be a very good tool for keeping your dog from engaging in bad looks with another dog. Any tool that can help prevent your dog from rehearsing a bad behaviour (ie lunging at another dog) is a good idea. If your dog is allowed (thru your own physical restrictions, not your fault) to lunge then that eventually becomes such an ingrained response that is becomes harder to retrain.
If you wish to abide by his rules and not use treats while taking his course then do so but there is nothing to stop you from using treats and rewards while working with your dog on your own time.
I also recommend spending some time studying dog body language. The key to preventing a lot of behaviours is to understand what is happening long before it gets to the point of lunging. It is possible the other dog is giving your dog the "stink eye" or other behaviours that is making your dog very uncomfortable. It is also possible that a lot of the behaviour you think is agressive is just over excitement. In any case the instructor should have split the two of you far enough away to reduce the tensions. A great resource on dog language is http://www.dogwise.com/ItemDetails.cfm?ID=DTB527
. In fact that website is an excellent source of some very good training books that you could explore.