My real job is as a geologist.
Yes, encountering rock at the bottom of your pond basin can be a significant problem. That could even be the cause of the leak in a karst topography. Some rotten limestone below your pond may have collapsed just a bit further, and even a tiny cave in of material into the void could have ruptured that clay seal at the base of your pond.
Even without karst voids, rocks in the bottom of the pond make it more difficult to compact the clay seal. If your contractor does encounter some rocks, he may need to compact a few inches of extra clay in those areas to give a little safety margin to your seal.
Another option for a "cleaner" doggie pond, might be to use an artificial pond liner. (There are plenty of threads on that option on Pond Boss, or you can just do a regular internet search.)
If you have construction equipment on site to repair your ponds, digging a smaller dog pond of an easy shape to use an artificial liner would probably not add that much construction expense. It will add the expense of the actual liner, plus the costs to properly "pin" your liner to the adjacent ground.
You could even utilize water from your other ponds to keep the dog pond full. You could also build a "dock" for the dogs to jump into their pond. You also need to build a means to exit the water for dogs and "accidental" humans. A steep pond liner makes it difficult to get out of that type of pond.
