Finally she's doing some real damage. This weapon isn't really meant for sustained damage but she can fudge the numbers a little and keep going. Still she'll fly down a little and try to gauge things so she does a critical injury without killing him.

If that doesn't seem possible... well he did threaten to kill her. That seems like enough of a reason to accept his death if it comes to that.

How exactly is she going to judge what is a critical but nonlethal injury for a transforming on-fire dragon with rapid regeneration?

The answer in a word is poorly. She's not really used to needing to judge people's health without health bars or other indicators so as long as he seems to be actively regenerating and trying to escape she'll keep firing. If he stops struggling she'll let up.

The nature of Lung's power, and the nature of Lung himself, is that he will never stop struggling.

Not to mention she's an unknown cape who attacked him without warning or any reason he knows of and is as far as he can tell still actively trying to kill him. As opposed to, you know, calling for him to surrender or something.

No, he will struggle and heal and grow stronger and OVERCOME.

...if she keeps firing, there will be a very brief window between Lung falling unconscious and Lung dying in which she could, in theory, stop firing.

Interfering in a fight with an unknown cape who is strong enough to defeat a highly evolved Lung by main force would be - unwise. If he interrupts or otherwise distracts her it could allow Lung to recover and get away, and he might be blamed even though it would be the result of her own incompetence.

He can't tranquilize Lung when he looks like that, because he doesn't know the right dosage. Too low would have no effect, and too high might kill him (and make Armsmaster responsible). The tranquilizer is meant to work when Lung is caught off-guard; if she can knock him out, he should transform back to human shape, and then Armsmaster will be able to capture him.

Protocol requires him to intervene to prevent the possible death of a person at the hands of a cape. Protocol doesn't quite admit that he need not intervene to prevent the possible death of Lung, but it does say he should not risk himself to intervene in a fight between a villain and a highly dangerous cape of unknown affiliation. 

Besides, he's already called for reinforcements, so he can ask Console for instructions and by the time they figure out how to respond this will probably be over.

Well, she should at least ask him to give up in the time she has. She'll pause firing briefly. "I'm not sure how to stop you without killing you and if I keep escalating it gets a lot more likely I hurt someone else by accident. Can you please give up?"

LUNG does not GIVE UP -

He's tracking Lung's regeneration and she could in fact knock out or kill him without escalating, but it's reasonable for other Tinkers to be cautious with the weapons they build. Lung is now weak enough and grounded enough that Armsmaster can finish the job by himself, which is all for the best.

"I can take it from here. Armsmaster, with the Protectorate," he adds in case she didn't take the basic step of studying the city's important capes before coming here.

A strike of his halberd cuts halfway through Lung's left wrist; that will keep his regeneration occupied for a few seconds. Next, a jolt of electricity to his head; it's different from the other Tinker's gun's effects and if Lung's power is partly adaptive as he suspects, it won't be ready to counter it.

grrrrr -

According to preset parameters, host now gives up. Or, more accurately, gives out.

—✳

And now he can be very efficiently tranquilized, restrained, and foamed for pickup. 

"Thank you for your assistance. You have helped apprehend a dangerous wanted criminal and helped the city and our fellow citizens," he recites. "He is now under arrest and will be safely held by the PRT and tried by the courts," which are certain to convict him but he is not supposed to say that.

"What is your cape name?"

She vanishes her weapon when it's clear they're not about to fight. "Oh, you're one of the hero people with the place out on the water! I made an appointment to talk to people at the building you have in the city tomorrow morning. I'm new here. And thanks! I wasn't sure if this was a 'to-the-death' kind of place or not and I didn't want to start off on the wrong foot."

Unobtrusively typing questions to Console while talking to someone is second nature by now.

"Lung does not have a Kill Order," he confirms. "Nobody currently in the city does. Of course, lethal force is permissible when defending yourself. You made an appointment using the name... Terrarian?"

"Yep! It's a fun name and a great yo-yo."

(The reinforcements materialize in the form of Velocity flickering briefly through the street.)

"How did this fight start?"

"Well, I was flying around and I noticed that this guy was on fire. He seemed to be fighting with a swarm of insects and when I asked him who he was fighting and what the fight was about he attacked me so I decided to fight back. Things escalated a bit from there when I couldn't use water to put out his fire."

"You didn't know who he was?" He shouldn't lecture her on being unprepared, she took down Lung with apparent ease and only stopped because she feared she'd kill him.

"I don't know of a local cape who creates or turns into a swarm of insects. Did you see another cape during the fight? Did the insects leave in an organized fashion?"

"After he started attacking me I wasn't paying as much attention to the insects. They did stop at some point but I'm not quite sure when. The only other people I saw besides you and the people who ran away when I got here were riding these big animals. They showed up part-way through but then they left before the end."

"Those were probably a local villain group called the Undersiders. Their member Hellhound can turn dogs into giant deformed beasts that they ride. They may have recruited a new member who can control bugs. The Undersiders have been reported operating in territory claimed by the ABB, and if they were not working for Lung then a conflict with him was likely." That's a simpler and therefore likelier explanation than four different sets of capes taking part in the same fight. The Undersiders attacked Lung, the fight attracted Terrarian, and she finished the job. Someone who saw the fight tipped off the PRT; Armsmaster was already on patrol and was dispatched. A clean report, with no loose ends.

"I have no more questions for you at this time. Lung's capture will be announced tomorrow and you will be credited. I may join the meeting you scheduled tomorrow morning, depending on who else is assigned to it. Do you have any more questions?"

(Two large black armored vans enter the street and disgorge a stream of people in opaque-faced black armor. They head straight for Lung and cover him in a grayish foaming substance dispensed from tanks on their backs.)

"I think that's just about everything. Oh, is it rude to ask someone what their powers are here?"

"Asking is fine" (or Armsmaster would be in a lot of trouble) "but most people won't answer, unless the powers are already a matter of public record. Having a secret power is very useful tactically, and knowing the details of someone's power could also let you deduce their vulnerabilities."

"Is your power publicly known? Also what was that guy's power? I couldn't really tell he had the fireballs and he seemed to keep growing as the fight went on."

"I'm a Tinker. I've built many different devices for different situations, so knowing that my power is building things doesn't tell you much. For example, I made a special tranquilizer that I used on Lung because ordinary ones wouldn't be powerful enough to shut down his regeneration."

"Lung is a pyrokinetic and a Changer - the longer he fights, the bigger and stronger he grows, eventually turning into a flying monster, like you've seen. He gains strength, durability, even faster regeneration, and he can both generate fire and is immune to it, but he doesn't gain any other new abilities. The recommended way to fight him is to take him out before he has time to ramp up, since most capes' power doesn't grow over time to match his."

"Oh good, that means I probably won't need to use The Last Prism very often. It's always more fun when I have lots of options instead of just always picking the strongest one. I guess it also means I need to be careful though. If he was supposed to be especially durable even to start with a lot of my weapons would seriously hurt people. Ah well, I guess I'll figure it out as I go."