I say the context would deter,one whether it was evil. Did the party get attacked in the bar and the bouncer joined the attackers and tried to kill the PCs? How do you think Conan would deal with that guy?
Which is irrelevant. Asking how Conan would deal with the guy doesn't shed any light onto whether it's a good act or evil, any more than asking how would Hitler deal with that guy.
Part of the point of the alignment system to start with is that the context is specifically not a consideration regarding what is good and what is evil. It's a Star Wars approach to good and evil (even though D&D alignments predate it) - killing Sith and the Empire is good. Killing Dooku who is defenseless and on his knees is evil.
A lawful evil society might condemn the killing because he should have been subject to the law (even if the law would have killed him anyway). In fact, both the fighter and the bouncer might be at risk for execution depending on the laws.
A neutral evil society wouldn't have any problem with the killing, nor would a chaotic evil one. Revenge might be a problem here, though.
None of this would turn the act into a good act. It is an evil act whether sanctioned by the society or not.
The only other alignment that could potentially justify such an act is chaotic neutral, but I don't think that even that would be sufficient here. I can accept a chaotic neutral character justifying killing a known criminal or threat to others, particularly a violent one, even when they have access to the law. But killing a bouncer who is doing his job, and there is no further risk (as far as we know) to the PCs or the people around them? Even that type of character I'd have difficulty rationalizing.
Which means that I guess that regardless of the professed alignment of the character, I would not only say that it's an evil act, but I
am questioning their actual alignment. At the very least, perhaps they are not of an evil alignment (yet), but they can't really be of a good alignment either. In fact, I'd probably say that at best they are chaotic neutral, but could be neutral evil or chaotic evil. They are certainly headed in that direction. I'm not sure that there's really any other alignment compatible with actions of that nature. I don't think it would be lawful evil.