Having such militarization be from before Kirk's era would still make sense. Starfleet pulling out of its war days and into a era of deep space exploration. Having the few battles mentions be part of a conflict with the Klingons also makes sense since the Romulans were basically unheard of for a century. That and the long conflict with the Klingons makes the idea not entirely improbable.
The concept of the Four Years War does work, or at least can work. War in Star Trek. At least with the Klingons rather than the Dominion, it was more a matter of Starfleet being outnumbered and partly unprepared. With the Dominion War, it seems more luck that the Federation survives at all.
Against the Klingons, the technology difference is not as great, and the back and forth would be easier to simulate based on real world context. With the Dominion, while it worked in Deep Space Nine, Divine Intervention was basically the only thing that saves the Federation. That and the roughly 70,000 light years between the Federation and Dominion without the wormhole.
Also the Four Years War seems to work as a mixed metaphor for the First World War with the lines going back and forth and the war not having a true end, but a delayed ending. The Federation finally beating the Klingons back to their own borders via technological improvements and possibly causing an internal power struggle in the Empire which then calls off the war to deal with internal matters. The Federation agrees since the Vulcans at least would want peace to prevail, and keep Starfleet from becoming the aggressors. I can't say if there would be reparations like at the end of the Great War. All we know is that the Klingons seems upset at the Federation over resources and feeling surrounded. The Organians stopping the second war from really starting thus pushing the two powers into a prolonged Cold War, ending only when the Klingons can no longer afford it.
One could speculate that the ending of their Cold War wasn't really the end of their conflict, but the events of Naranda III with the Enterprise-C defending Klingons to the death made the final impression on the Empire about the Federation. Though that peace lasted until the Dominion came and soiled the treaties, though that was ultimately resolved with a far better alliance to combat the Dominion in the end.
