You need to increase your animal affinity enough. It is a hidden stat that is easiest to raise by submitting to wolves in the forest.
Not sure if you're still looking for a solution but I did find it. Either once your bitch fame is high enough and/or you get the dog (not sure which one, I did both) an option opens up when having fun in bareshade to have fun with the stray dog, you should only need to to it once to be able to train the one you have at home.