See, I don't think we can really answer the question "does a mortal using a sponsor's magic to break the Laws assign the Lawbreaker power to that mortal?" because we don't know exactly what Lawbreaker represents, and how it comes into being.
If it is purely a stain on one's soul, 100% internal to the person using magic, then any mortal who breaks a Law purely with sponsored magic should get Lawbreaker. The reason I don't think this is the case is precisely because subduing someone with magic and then shooting them in the head is not a gray area at all, the Wardens do it regularly. You've still used magic to kill, it just wasn't the actual instrument you used to deliver the coup de grace.
If Harry is right, and magic really is the essential force of life and happiness and puppies, then breaking the Laws is perverting the nature of magic, and Lawbreaker is the backlash of that act. It's an outside force damaging the soul of the user. In this case I do not think that a mortal using purely sponsored magic would become a Lawbreaker, because the power in question is not the power of life, but the power of some other entity, generally one of the Faerie courts (since most other types of sponsored magic require the user to have magical power of their own).
The gun analogy doesn't apply since it isn't the act of killing that is problematic, but the act of killing with magic (or breaking one of the other Laws). Unlike the "guns don't kill people" argument in this case the instrument is absolutely the issue.