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I read up more about the situation, and now it seems Moscow either thought Japan won't dare to declare war, or they were very confident they can beat them.
Japan sought peaceful resolution, tried to barter a deal, but Moscow snubbed them and pushed them enough so they did not see other choice but war. While in Korea a pro-Russian government reigned, the Russians themselves had no business south of the Yalu. But through a chap called Bezobrazov, who acted as a private entrepreneur - but backed by directly the Czar, had Russian soldiers as muscle -; pushed the Japs further with activities along the Yalu on both sides.
So as for if Japan could have tried some other way to dislodge Russia, I'm not sure. Russia exploited it's status along with the other European powers to gain what it could from China, but Japan had no status (until they beat the Tsardom).