I live on the HH Palms and when there is no golf I have played fetch with my dog in the rough. I can't believe that is a monumental problem. It's very dangerous to do anything in front by the race track. (road). I certainly wouldn't let him go on the green or poop on the course.. It it did happen I would clean it up immediately. (They are usually unscheduled.) If you want to see a green full of crap put a bunch of geese on it. Unbelievable!!
So come on, let my puppy have a little fun and me too! Life is a very short project so lighten up! It's not good for your heart to be grumpy. I've seen golfers do more damage to greens than a dog could do in a week.
He's my buddy!