Acknowledging that they are obese is stating a fact. Claiming that they wouldn't or couldn't take good care of their dog because of their obesity is fat shaming. They're not 600+lbs-can't-leave-the-bed obese.
So obviously that means he couldn't possibly ever exercise, because people who lose weight just wake up suddenly skinny and are never still fat at any of the in-between points during which they still get constantly shit on for existing in a shape society disapproves of.
These people will just use their mobility scooter while the dog has to run for it’s life while being leeshed to the scooter. Seen that scenario plenty of times in my town.