If I had to use putty it would be Famowood. The putty is already pre colored. And holds a stain quite well. If it's just chew marks with nothing missing you use the heat gun or wet a towel and use an iron on the highest setting. It will raise the wood. Or leave the stock as is and it will have character. Every time you use it it will remind you of your dog. I have an old duck call from the 60's that my dog chewed the barrel down about 3 inches. I cut it down to make the end even and still use it today. Brings back great memories of a good dog.
Since you are going to refinish the stock you will sand it. Collect the sand dust and mix some white glue with some of the wood dust into a thick paste and fill the holes or dents with it. When iot dries it should be a close match to the color of the stock.
When you are up to your butt in alligators, it is hard to remember your intent was to drain the swamp. Stay focused!