Wiseblood
Adventurer
The problem with 2d6 is that it isn't coming from a weapon, it is coming from a natural attack that cannot be disarmed.
Moreover, it is specifically narratively the bite of a dog. The bite of a wolf deals 2d4 damage and can cause knock-down. And wolves generally have far greater biting strength than the vast majority of dogs. Wolfs biting strength is between 400 PSI. The only dog that outdoes it is the Mastiff at 550 PSI, but the next highest is Rottweilers with around 350 and literally all others dogs with less than 250. Since we can be pretty sure this class doesn't universally represent Mastiffs or even Rottweilers, we can surmise that 2d4 would really be much higher damage than can be expected.
A Worg's bite causes 2d6 damage, but a Worg is a large sized dog-- i.e. it is over 8' long and weighs over 500 pounds.
So that gives us the range of what is appropriate.
Stealing the monk's damage progression seems like a way to go, presuming you give the dog the same ability to make an attack as a bonus action and something akin to the martial arts ability to spend "ki" to turn that into 2 attacks as a bonus action (or dodge or disengage if you choose).
True you can't be disarmed. You also won't find magic teeth. Bite pressure is strength. I wonder why you are okay with the dog being able to bite faster but not okay with bite for more damage. D6 plus stat two times is greater than 2d6 plus stat once and two attacks will take more time at the table while increasing the chances for a crit.
Pehaps a conditional requirement like sneak attack or a limited resource like superiority dice?