I think killing every carp you catch is a waste of time. How many have you caught, 2? That won't make a difference in the population, at all. Also, I think throwing them on shore or slitting their air bladder is unethical. That reminds me of guys in Florida catching catfish and throwing them on the pier to die because they considered them a "trash fish", they didn't make a difference on the catfish population. It just made them look like bad sportsmen.
But, if it's Asian Carp, keep as many as you catch and have a fish fry. They will destroy a lake. In fact that happened to a lake I fish quite often.