I'm really undecided. I really like your shot, just the way it is. It has beautiful detail and lighting. The background is nice and I wouldn't have even given the circle a second thought unless someone mentioned it. But then, I don't notice a lot until people mention it, and then I learn from those comments. Hawkeye's is very nice too, although to me it looks a little fake and unnatural. I guess my personal preference would be yours -- but that's just the opinion of an amateur!
Like most things with Photoshop there are about 10 ways to do the same thing. I tried a couple of different ways before I settled on this that gave me the best bang for the buck. By spending a little more time with the brush close up you can prevent losing the glint off the whiskers while still removing the bright spot.