As someone who has had a torn labrum repaired, I would be surprised if you feel like picking up any bow for a couple of months after surgery, even if you were to try to draw using your mouth. It is interesting that we use almost every part of our bodies when we shoot. You will be in a sling for some time and then slowly gain strength. I had my shoulder operated on during the first part of December 2005 and was able to do some not-too-heavy lifting when I moved office buildings in February 2006, but it was difficult and hurt. Good luck with the surgery. Hope you recover quickly. If I were in your shoes, I would try to find the lightest weight bow that you think is adequate for your hunting and then not plan to hunt until November, assuming you even are able to do so then. You and your physical therapist will know how you are doing as you recover.
Allan
Allan