Killing prisoners is not always an evil act.
If bandits are burning farms with people inside them and the party kills some and captures others. The bandits are questioned as to who they are workin for and why. There isn’t good justification for their banditry (Robin Hood) then it’s not an evil act to dispense justice and kill them. I’m not sure where this idea that killing a prisoner is evil came from.
Legally speaking, it came from the Geneva Convention at the end of World War I, which specifically bans the use of several uses of what was collectively viewed as horrific acts of dishonor and disrespect on the warfront, including things like biological/chemical weapons, killing prisoners that have surrendered, and certain forms of torture. But the idea of treating prisoners with respect and keeping your word definitely predates it conceptually and throughout most of history those who treated their prisoners poorly could largely expect the same treatment if captured or suffered severe issues when trying to be diplomatic as word got around quickly that they were not to be trusted. Hence why in my original post I suggest that if his players continue such behavior the logical result should be them slowly being ostracized by the very society they deem "civilized" if not outright hunted down by it.
Even in a highly regulated society vigilanteism isn’t necessarily evil. In a society without national police or legal system it would be normal. Remember most law enforcement was conducted by private citizens until the late 18th C or even later. Most serious crimes were punishable by death and there were over 200 crimes on the ‘bloody code’.
Such societies also still largely distrusted those known as vigilantes widely as any who tried to wield the law without proper authority to do so were often quick to become the very bandits they were supposedly there to keep at bay. Also, the DM has given us no context on if his world has no such law enforcement/standing army or what his legal system is for the specific area of where these events took place. Even if we look at real history not every nation was the American Wild West, and if anything such lawlessness was almost the exception to the rule in terms of law enforcement; there is a reason why the image of the American Wild West and Americans as "cowboys" are largely viewed as incredibly "foreign" or "chaotic" by much of the world.
That all aside, even if we remove the good vs evil aspect of the argument over whether killing a criminal is wrong (It is still an evil act unless there is no means of bringing them to trial, the neutral response is to only bring them back if it is reasonable and the evil one is to kill them because "it's easiest"), we still have the law vs chaos arguement. If such systems exist, the concept of bringing them back alive if possible is the lawful response, as is honoring your word that one would be let go. Said player in question was Lawful Good, and it's perfectly reasonable they'd be upset the group has effectively made them a liar against their will, paladin or not. I still maintain my previous statements: have the player reroll and in doing so, inform the local government or his religious order that the party is in effect a group of weyward wandering murderers. If wandering adventuring bands are legally right to declare that bandits are wholesale evil and deserving of murder on the spot, so too could an order of priests dedicated to a god of Justice/Law to the very same group of adventurers using the same logic. After all they ARE murderers, even if those they murdered were criminals.
Again, I take no issue with evil PCs (hell I've played in such games and they were loads of fun). But they should realize and own that they ARE in fact Evil. And logically face the consequences for their actions within the respective world.