Personally, if I was another passenger on one of your flights, I would have rather your dog missed its connection and arrived a few hours later than causing all 300 other passengers to wait 15 minutes so that they can make sure that your dog made the flight.
My flight from ATL to SEA in October was diverted to another departure gate in ATL after waiting for a 757 to pull back. The 757 captain would not leave until a dog that was missing was tracked down and put on the flight. We waited in the terminal for about 30 minutes before DL gave up and sent our arriving plane to another gate--the 757 didn't leave until it was about an hour and a half late. The change in equipment (our arriving plane ended up at a different gate than we were sent to) we then experienced led to 2 more gate changes and a delay of about 3.5 hours for our flight to SEA--in fact the next flight to SEA left before ours (and yes, I tried to get on it as standby but it was full).