Well OTL Russia nearly won because Japan couldn't sustain the war, a better wake up call cannot be found.
Of course, one can (I'm playing the Devil's advocate there) argue that if NII or rather NN and Witte did not freak out and kept ignoring the domestic disturbances, then Japan may ask for peace because it could not continue the fighting due to the absence of money. An important part of such an argument would be that assessment of the military situation given by NN was bogus. On one hand he acknowledged that situation at the front is stable and the noticeable Japanese success in a near future is unlikely. The "panicky part" of his argument was that with a complete freedom of the naval operations Japanese can attack any of the Russian ports on the Pacific with landing the troops. IMO, taken on its own, this argument was not a serious one. They would land where? Okhotsk or Petropavlovsk-on-Kamchatka? Be my guest and good luck with supplying your troops because from each of these ports you can go nowhere. Sakhalin? Russian Empire did not know what to do with it and Japan had fishing concessions on its Southern part, anyway. Vladivostok was the only place worth attacking but it is rather difficult to get to, not such a big problem to defend (as the only point of defense) and, again, it gives nothing because Russian trade on the Pacific is almost non-existent and opening a new front is stretching the limited Japanese resources even more.
Witte's part of the argument was seemingly more solid. He argued that a need of the further mobilization may cause the greater unrest and the measures needed to crush it may cause a negative PR in France and made the middle-class French more reluctant to sign to the new Russian loans. However, one could run 16 wheel truck through his logic. To start with, if NN stated that the existing forces on the Far East (which were a fraction of the existing army) are adequate for containing the Japanese, talk about the needed massive mobilization sounds not convincing: just send few more of the existing divisions (IIRC, during the talks in Portsmouth the news about arrival of two fresh Russian divisions to the front forced Japanese delegation to agree to the peace without reparations). Then, while a French rentier may be somewhat irritated with the information about suppressed revolts of the Russian peasants, I suspect that in a choice between compassion to these peasants and a well-paying secure investments the peasants are not going to be the winners.
Then, of course, it could be argued by NII (at least as a tool of a public demagoguery) that if the French will prove to be too obstinate and anti-Russian, the Franco-Russian mutual defense treaty may be easily replaced by the Russian-German one. Anyway, if the uprisings are crushed, it is just a matter of a reasonably short time when the greed is going to win over the sensibilities (what's done, is already done and we are not going to help anybody by missing a good investment opportunity).
I may be wrong but IMO "the mindset of the day" was to get rich fast and to think about the consequences later.But let's go back to your premise that Russia didn't have a good enough reason. I actually agree with you, especially you could have divided it up with the Japanese as was suggested before in this timeline by @jessicajsscjs, but this time I've quoted from Wiki (as unreliable as that is)
Dump the idiotic demand for the bufferzone and give Korea to the Japanese. No war needed.
Edit: Ps, I still think the mindset of the day makes a non ASB Nii go for the war. But if you can argue for no war (which makes sense) then I want my Manchurian resources 😉
Russian top aristocracy, including the imperial family was getting short of money and "Bezobrazov's adventure" looked as an easy way to get money fast and effortlessly. The "monkeys" are not a danger so let's keep violating our own international agreements pissing off not just Japan and the GB and US as well. Even Witte got overly-excited with the ideas of the Chinese and Korean markets as the way to improve the Russian finances and did not bother to consider the geopolitical consequences. Or you can say that this was (besides the pure greed) a typical Russian approach: extensive rather than intensive way of doing things. Expansion into Korea (which was Japanese zone of "interests" by the existing agreements) could make sense if Russia already managed to loot all existing resources of Manchuria and fully exploited resources of the Russian Far East. But this was far from being the case. Take, for example, Sakhalin: the oil on the Northern part had been discovered in the 1880s and there were numerous geological parties and even some commercial interests but the first extraction started only in 1908 by "German Chinese company" and the first Russian company started extraction only in 1910, simultaneously with the British company, but serious extraction started only in 1921 by Japan and continued as a concession after in 1924 the Japanese troops left the island. If instead of getting into Korea there was a serious interest in the development, Russia would be able to sell Sakhalin oil to Japan (as happened later) or just keep getting concession money.
Back to the subject, it would not take ASBs to make NII more like his father in the terms of being able to keep things (and especially the court camarilla) under control and to look at a greater picture.