Need a sensor to make ball roll away from pet but which sensor?
Perhaps an RFID tag on your dogs collar and a reader inside the ball will work.
You will just have to find a reader and tag combination that can work from the range you need, I think some RFID can work from over 1ft.
EDIT: You will need active tags (battery powered) to get over a foot range.

PIR sensors detect heat given off by humans and animals, so they make work through the ball. (not totally sure though).

Edit: i think PIR that is capable of detecting small animals may be your best bet you would just have to calibrate the range of detection some how