What happened to make your dog sore? Was he running about and the leg suddenly got sore, or have there been any problems with that leg in the past?
If it happened suddenly, the injury will either be a strain/sprain or cruciate ligament damage. Bichon frises are a breed more prone to getting cruciate problems, so your vet is right - it is likely this is the cause of your pet's problems.
There shouldn't be any reason not to operate because of size - some surgical techniques are most successful in smaller dogs. Leaving the knee joint unstabilised may also cause further problems, e.g. arthritis. However, there are treatments that don't involve surgery - things like stopping the knee from moving and physiotherapy; maybe this is what your vet had in ming when he said he didn't want to operate?
In any case, tell your vet that you are worried about your dog, and he should listen to you. If you really want surgery, ask him to pass you on to another vet who will operate, or get him to refer you to an orthopaedic specialist who will do it. Failing this, go and find another vet!