Successful breeding of Pets?

Just a guess, but I think the 'be nice to' interaction is really not working, so there's no way (except the mod described above) to work on dogs' relationships except hanging out with them a lot and hoping for best. It's also clear that the dogs need some level of acquaintance, maybe even friendship, before mating, so that's a deadlock in many situations.

If you have your heart set on breeding, I'd recommend creating the intended breeding couple together with their owner, maybe even as puppies and let them grow up together to maximize their friendship.