No deep disagreement there, and no pressure!But you're not going to convince me of it

It is just that every one of your objections just reads "this is not solid proof", rather than "this proves there was no war" or even "with this evidence, war is less likely than no war".
No need for any single one of the wars to take longer than a few weeks, really. Remember how Larry Niven did it originally? He first postulated four wars, then let others play in his sandbox and postulate a war with four back-to-back attacks (necessarily quite separate, as the Kzinti there were strictly sublight), and then established that this four-parter had been but the first war. In Trek, the four-parter could well be the four wars, all taking place within three or four years. It is a motif of the adventure here that the ratcats just plain don't learn their lessons, after all!The problem with this theory is that Sulu specifically stated that the last Kzinti war took place 200 years before TAS. That implies that the three previous wars took place even earlier
It's possible to argue that 150 years would be rounded to 200, even through the opposite is not plausible, and even though rounding up 149 years to 200 is even less likely (but can happen, if the person first mentally rounds to 150 and then further rounds up). Sulu here would have been rounding as far up as he can - "you were completely defeated 200 years ago already!" would be somewhat weakened if it went "you were completely defeated 172 years ago already!" because the point is to convince the cats that they were defeated long ago.
Thankfully, Sulu doesn't claim that the wars would have been "over 200 years" ago, merely "200 years" ago.
The basic concept of Earth defeating space aliens early on is probably valid enough, as we already have to believe in Earth quickly creating warp-capable deep space ships such as the Valiant. If those were created by warp-uprating preexisting ships, like the canonically unseen Greg Jein model would suggest, then a vast Earth warfleet from WWIII could have been warp-uprated the same way. Or then Earth would triumph with sublight vessels, sort of how the unseaworthy Monitor could fight "real" warships to a standstill at the very least.
Timo Saloniemi