GreenTengu
Adventurer
I don't think it's a matter of justifying the PC's actions. For me, it's a matter of innocent until proven guilty; remember that in the case of murder you must establish guilt beyond a shadow of a doubt. The OP's description of events left a lot out. We don't know whether they were in a frontier town or the heart of civilization. We don't know who attacked first, nor have we been made aware of who used lethal force first. We don't know how the fight got started.
What we do know is that there was a bar fight. The bouncer engaged the PC with a sword at some point during this brawl. A few moments after the bouncer surrendered the PC killed him.
Were there extenuating circumstances? We don't know. I'm not a lawyer, but if this PC were to be tried in a modern day court (presumably in absentia, since we are lacking his testimony of events) given only the evidence that we have been given, I suspect that more likely than not that you'd end up with a hung jury. The specifics of an event matter, and in this case all we can do is speculate (or make assumptions) about the particulars.
Now, as I've said previously, I don't believe that morality and legality are directly correlated. However, murder is an act where the two tend to be in line with each other. Obviously, just because the PC isn't found guilty in a court of law doesn't mean that the PC is free of moral responsibility, but it does suggest that there exists at least the possibility for this to have been a non-evil act.
Personally I believe that any killing of another sentient being, whether you can justify it or not, is evil. It might be considered a necessary evil, such as when killing to protect innocents, but that doesn't make it good in my book. Heck, I think that most killing of even non-sentient living creatures is evil. Squashing that spider that was minding its own business? Evil. Not your soul will burn in hell for all eternity evil, but still evil. I catch and release whenever I can for that reason. That doesn't however make mine an objective truth. At best, it is my truth, purely subjective. If an objective truth of good and evil exists, the definition of it is presumably the domain of an unknowable higher power.
Let's see here...
Let's say the PCs were as much in the "right" as possible.
The PCs were at a bank and someone attempts to rob it. The PCs attack the robbers and there is a big fight. The security guard comes running in, pulls out his gun and tells everyone to stand the hell down because at that point he doesn't know who are the robbers and who are the vigilantes.
The PCs then beat up the guard too and when they have him down and surrendering, they shoot him in the head.
And here you are playing defense lawyer for the person who shot the security guard in the head for doing his job-- claiming extenuating circumstances and using special pleading to try to justify it.
What I am saying is that if it was an NPC Orc who had taken that action, you would not hesitate to label him guilty of being evil and enact the death penalty.
Killing the law enforcement who are trying to keep the peace in an act of vengeance or power just because you were, potentially mislabeled, a miscreant is not just Chaotic-- it is evil. A Chaotic good character may have chosen not to fight at all or at least fight only until the authority trying to intervene can be spoken to and negotiated with.
And, yes, a bank guard and a bouncer are in fact equivalent-- both are private security hired to protect a place where crimes are likely to occur. Even if you have all the respect in the world for one institution and none for the other, the jobs are functionally and ethically equivalent.