*coughcough* hannahascher *coughhack*
Nope. That wasn't an
execution, that was a combat situation, and he tried his best to talk her down and when that didn't work, he very specifically used her own magic against her to take her out. Right on the razor's edge of the First Law, but he is a Warden, after all.
I strongly suspect that at some point Harry will find himself in a situation where he basically has to
execute a helpless warlock, using a gun or a blade, in his capacity as a Warden. Not combat, just cold-blooded killing of a human being who is powerless to fight back or run or otherwise do anything about it.
Take out the gun, point it at the helpless, immobilized warlock's hooded head, and pull that trigger. I suspect Harry would prefer to do it with a gun, if he has to do it.