Everything about your reasoning is fundamentally screwed up. We're talking about the possibility of having two confirmed town players. Except to the extent that confirmed town players are always a target, the Dog in the Know is simply not "a role to go for" to the mafia, because it has no particular features other than being confirmed town. Once the actual information of the player that player is in the know about is out, there's no mechanical difference between the Dog in the Know and anyone else. So your whole thesis is wrong and frankly rather scummy.
Simply put, having two confirmed town forces the mafia to let us go into day 2 with at least one confirmed town, which is a strong benefit; and the doctor can protect one, forcing at least a 50/50 shot of having two confirmed town on day 2. That's a big deal given the existence of only two mafia. To make matters "worse"... by your own acknowledgement, the cop is of limited value in this setup as a cop. On the other hand, the cop is of greater value as a definite town role. In the event that the cop is neither the Dog in the Know nor the target of that knowledge, that would mean three confirmed town if the cop claims. Four if the doctor is also neither of them - so best case scenario the mafia is narrowed down to two out of three. Now, this obviously forces the mafia to counterclaim, but even then, that can only involve up to two players, which means, in many of the possible arrangements, the game is still already solved.
For all intents and purposes, the only role that's a significant target to the mafia in and of itself is the doctor. The cop is fairly weak under the circumstances, having to hit the henchman round 1 to have any real impact. Certainly, if either the Dog in the Know or his knowledge-target is also the doctor, I would recommend keeping the doctor part secret and quietly protecting the other of the two. If both of those players are vanilla town, then a full claim will confine the mafia nearly completely. It's also possible that they are exactly the doctor and cop, which would be the least useful of the outcomes for town, because mafia can simply choose not to counter-claim, and are then only restricted to being 2/5... but that's still pretty good compared to 2/7. It also obviously follows that a doctor or cop who isn't the Dog in the Know should also want the Dog in the Know to be revealed: it guarantees that the actual power role will live longer.
Or basically, in short, another way to think about it is: the mafia will always know they're killing a town player at night. The information that a specific player is town, then, doesn't benefit them in any way. The only value that the Dog in the Know role has over others is exactly that information, so revealing it is better for town.