Note that in 'Patterns Of Force', Kirk was yet again trying to redress the interference of another. John Gill, rememebr?
Also, the messed-with culture was reaching into space with the stated intention of exterminating the people on the next planet over. Things had gone too far for the Feds to simply extract their man, back off and hope for the best.
Think about the strategic ramifications as well. A runaway culture that embraced Nazism, had space travel, MIGHT have had the basics of warp drive (never mentioned, but Kirk had no way of knowing just how much Gill told them) and
was on the hunt for Lebensraum AND Concentration Camp fodder.
Great, just what the Galaxy needed.
All in all, the situation was definitely beyond bleating about the sanctity of the Prime Directive. How many other cultures could those guys have messed with (read: enslave or exterminate) once they got going? What about the right to non-interference of those other people?
You cause the mess, you gotta try to clean it up - or, at the very least, stop it from becoming any worse.
If anything, Kirk was restrained. Enterprise could have stopped the entire invasion cold. I clearly recall that one Zeon pressed him to do that, but Kirk refused out of concern for the casualties that would certainly be inflicted.