I dont see how you could have the USSR go to war with Japan with out American intervention .. The United States is Japans military. Something like that would lead to a full on east west confrontation, Unless the United States has gone isolationist and just given up.
Well, it's a relatively limited naval and aerial engagement until the last phase of the war; also WMDs were never used. Moscow never intended to invade Japan and are out for a status quo ante bellum and some prestige. If it went further, however, with Soviet boots on Japanese soil or the USSR using nukes, things would be a bit different.
The war also wasn't intentional, or at least many could argue it wasn't. Many world leaders and US politicians could point at Japan and the 'aggressive' behaviour and 'violation of protocol' of the F-15 pilots in what was a mere accidental violation of Japanese airspace, thus justifying the Soviets opening fire. Mind you, this isn't my personal opinion, but it's how people ITTL could perceive this.
Lastly, the US and the USSR don't want WW III over this.
also i would think that kim might take the moment to walk south..
Methinks Beijing and Moscow would disapprove. In this case, US forces would be directly under attack from enemy (North Korean) forces which is sure to lead to some kind of response. Widening the East Asian crisis is just dumb; all sides know that.