I think that it's a chaotic act, as were all the other killings, but I voted "other" because I wanted to emphasise that it doesn't change his alignment.
"Lawful" doesn't mean "law-abiding"
per se. "Lawful" means that the character believes the best society is one that is ordered and structured so as to provide stability - lawful neutral characters tend to think of this as an end in itself, while lawful good characters tend to believe that an ordered society will produce genuinely positive effects in its citizens' lives - they will be safer from harm, they will be provided for in times of hardship,
et cetera. In other words, they believe that the aims of Good will be best served by an ordered society. Lawful evil characters, on the other hand, support social order and legal codes so that they can be reliably manipulated for the characters' benefit or that of those they care for.
Laws, in and of themselves, do not necessarily create the kind of stable and reliable social order that lawful characters believe is best. The laws proclaimed by a chaotic good king are not likely to tend towards the establishment of social structure! Even without bringing up the conflict likely to arise between a lawful good character's beliefs and the legal code of a lawful evil nation, you can have outright conflict and incompatibility between "laws" and "lawful alignment".
It's clear to me, at least, that any definition of "lawful" which relies upon the idea of following local laws is deeply flawed. That said, what
is true is that lawful characters will tend to prefer to disrupt the structures of any society in which they find themselves as little as possible, because even when they fundamentally disagree with the specific structures of a society they won't want to exacerbate the problem by their own actions.
For example, a lawful good character in a neutral evil society will probably try to work for the establishment of laws or traditions which correct the abuses of the society and increase the safety and happiness of as many people as possible. They're also likely to have their hearts broken when lawful evil people manipulate the new systems.
In this specific case, the fighter's being in one country or the other is irrelevant. In the first, lawless nation, his actions represent the individual efforts of one man - he was not attempting to establish laws or traditions or structures by which paedophiles could be punished, but taking justice into his own hands and eliminating the threat to children himself. If he is otherwise the kind of person who believes in social order, then it represents a slight inconsistency in his character for him to take such direct personal action.
In a lawful good country, nothing changes. He's not working within the social order - he's neither obeying the laws nor even assisting the authorities in tracking paedophiles down. He is, instead, acting on his own "authority" as a vigilante - he may otherwise be a supporter of social order and structure, but this compulsion to act alone against such wrongdoers remains a chaotic component of his personality.
(I'm skeptical of any description of a "lawful" character that relies upon terms like "a strict, disciplined life, and an ordered, rational way of thinking". These are things with no direct bearing on one's opinions on the best way for society to be -
if the fighter expected and encouraged similar discipline and rationality in society as a whole, that would be an indication of lawful alignment, but a structured "inner life" says little about one's social views.)
If the fighter, while in the lawless country, had attempted to build some kind of structure to prevent child molestation - even a semi-formal agreement between regional "authorities" (be they local warlords, strongmen, what have you) - that would be more in keeping with a lawful alignment, as well as a good first step for increasing social order and structure. If you can get the strong personalities who control their local areas to agree that something needs to be done about child molestation, you can presumably get them to agree on other things, and start to build some kind of nationwide council which can establish laws, traditions, and other apparatus of an ordered nation. One analogue would be a certain kind of romanticised image of a Wild West sheriff, one who comes into town and tries to impose order onto lawlessness and encourage other towns to do the same.
In the lawful good country, a more appropriate course of action would have been to work with the authorities in some capacity - either through anonymous information, or through openly offering one's services as either a member of the "police force" or in some related "freelance" capacity. A modern analogue would be the formation of a pressure group attempting to influence the authorities into devoting more resources to the issue - i.e., working within the established structures of society.