We have done the same on many multiple occasions,... that being ... bringing our dogs with us to an eating establishment but sitting outside the building.
We've never been told "no" when sitting outside. However, if their outside eating area is enclosed (like a fence or a wall and you access the area through a gate from the outside or have to enter that area by passing through the building first, ... then "that" is pretty much the same as sitting inside the building.
If you want a good reaction from people .... well ... many years ago we had a pet chicken. She was leash trained, rode with us in the car, on bicycles, and went swimming with us. She was a remarkable chicken. The kids wanted to go to Dairy Queen. They had outside seating. We took the chicken with us.
Well, maybe it wasn't so much we had the chicken with us, as much as everyone was bewildered that she enjoyed Dairy Queen Ice Cream (vanilla).