The dji drone have a tracking function that may work for dogs.
The image processing based tracking seem to be geared for larger objects, that move a more predictable path, in plain view. The two videos I have been able to find from automatically tracking drones lost the dogs repeatedly when they changed direction suddenly in the grass\brush.
There is a drone that uses a TV remote sized GPS tracker, but it does not appear to be released for sale yet. The promotional videos for it look promising, but the remote is not water proof. Maybe a zip lock bag and some duct tape
Spent a couple of hours this evening researching and exchanging ideals with a guy that uses drones professionally, in his photography business. His opinion is that the flight processing\nav software is nowhere near being able to track a seemingly random target like a bird dog. I also found out that due to FAA regulations, none of the drone manufactures will provide APIs or other means of interfacing their software to a location signal from another source. So even if we can hack the Garmin signal (very do able), we cannot get the data to the Drone.
Seems like another whiskey fueled idea shut down by reality.