I'm from Maine…Portland, not Bar Harbor...but here are a couple of ideas.
There is the Dog & Pony Tavern...you can take your dog with you to have a drink...don't know if anyone has actually brought a pony there ;). There are the carriage trails in Acadia National Park which are amazing...not sure what restrictions there are, but I used to take my boyfriend's Jack Russell hiking there...Apatche loved it!
And there is a monthly dog newspaper from Bar Harbor that might have some other ideas...Northeast Dog News
I'm sure if you ask the inn keeper they can give you some other info…there must be a dog park as the town is super friendly to dogs! Also, it is possible that in September some beaches might be open to dogs.