No offense, but he can't use those firearms if he's unconscious.
However, I could see her arguing that she wasn't sure she had given him a concussion with those punches (which is necessary to keep the victim from just getting up again three seconds later), and it was proven that Coil had moles inside the PRT building. She could conceivably argue that if she didn't kill him, she was risking him getting back up again and going through with his plan, especially since as far as she knew he could conceivably have made sure the armed security guards that picked him up were in fact his moles.
I would compare this situation to the death of
Jacques Mesrine. This was a bank robber who was legendary for escaping from prison several times, disguising himself expertly, and robbing banks with such unbridled impunity that he became seen by the public as a "Robin Hood" figure. The gendarme knew they couldn't hold him, and knew he would never surrender if he could get away with it, so they just boxed in his car and had a bunch of police marksmen go full Bonnie and Clyde on him in an extrajudicial ambush killing. After shooting him 15 times, they
finished him off with a point-blank shot from a pistol. No-one was ever charged for this obvious case of police assassination, despite his innocent mistress, also in the car,
losing an eye, being shot in the arm and her poodle killed in the barrage.
This is a similar situation, albeit only between two people. Piggot doesn't know if Coil has any more backup plans or hidden agents in the building. She also knows Coil has a tendency to go into any situation
extremely prepared, with dozens if not hundreds of contingencies in place and contingencies for those contingencies. And seeing as Coil just marched into her office with two guns, it's obvious from her perspective that he intended to kill her. And now that she finally has the option to nip all of those contingencies in the bud, and isn't sure he won't just get up again at the drop of a hat? Yes, I could totally see her getting away with the killing under those circumstances. On top of that, she's a high-ranking and unpowered member of the country's superhero division and he's arguably one of the most dangerous villains in the city, and a Parahuman to boot.
That's three different tried-and-true excuses right there: 1) The Bonnie and Clyde defense (he was too dangerous to be left alive. Police tend not to be prosecuted in these situations), 2) Normal human vs dangerous parahuman, and 3) The perpetrator was both an authority figure and a cop.
Also, Earth Bet has thirty years of cultural divergence on us. Who's to say there's not a law or regulation somewhere that's meant to excuse the PRT in this exact situation? Maybe the Bonnie and Clyde defense really
is enshrined into law there, without even a kill order being necessary. Of course, it's up to Ack how he'll deal with this, so I can't state that it's there for sure, but it's his story and I trust him to wrap things up neatly.