You know though... the sheer amount of people looking to justify any PC's actions to dodge labeling the PC "evil"....
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.
The alignment system, of course, does permit killing within the definition of good, so let's look at the definitions:
Lawful good creatures can be counted on to do the right things as expected by society. I think we can agree that unless this society has some really unusual norms, this was not a LG act.
Neutral good folk do the best they can to help others according to their needs. While you could perhaps argue that due to some unknown variable the bouncer needed to be put down, this was probably not a NG act.
Chaotic good creatures act as their conscience directs, with little regard for what others expect. Was the PC acting according to his conscience? We really don't have enough information for that. However, if he honestly believed the bouncer to be a threat to others, this could arguably have been a CG act.
As such, I honestly don't think that it's as simple as declaring the PC's act as evil. There is room for doubt in my mind, particularly within the scope of the alignment system (as opposed to real world morality). As I enjoy telling my players, non-evil does not mean nice. Jack Bauer, from the TV show 24, could arguably fit the definition of CG as defined above and he wouldn't be above killing an enemy who has surrendered if he had reason to do so.
The more one delves into it, the more the alignment system in just utter crap and fails on the most basic of levels. Or, in the very least, it is wildly misapplied within the structure of the world so as to be subjective from the PC's point of view rather than some omniscient neutral entity classifying things judicially.
In my opinion, the alignment system is about as successful in describing a real person's psychology as hit points are at describing a real person's health. Which is to say, they're both extreme simplifications of real world concepts that are reasonably useful as gamist constructs, but they are hardly realistic. Unless you want injuries in your game to be more simulationist, hp suffice. Similarly, unless your intent is to undertake serious philosophical exploration of ethics and morality, the alignment system is sufficient. In the classic style of game where you slay the dragon, save the princess, and earn treasure and acclaim, you don't need more. As long as you don't expect them to be more than they are, they work fine.