Nah, Harry doesn't want to help the Council at all. The only times he ever has is because he was strong-armed by someone (Mab calling in a favour in SK, Luccio conscripting him in DB, Morgan going to him in TC). On the other hand, Harry keeps expecting them to do things like fight wars for him while he openly defies, mocks, and belittles them.
Then he gets self-righteous about how the Council distrust and dislike him and how dare they, he knows that he's a good guy, that should be good enough for them.
Harry is a hypocrite that keeps giving/showing more trust and openness to literal Monsters then he ever has to the Council, and then whines because they don't trust him.