The Japanese would be doing this without American backing. And the Soviets still had nukes.
The USSR was in a state of collapse, but few things unite Russian nationalists, communists, national bolsheviks, and even greens (because of the whale hunt) like getting really chesty about Russian dominion over the Kurils. I think somewhat recently I posted a video of a protest in Moscow where all of these groups got together to condemn Putin for talking to Abe about the issue.
In addition, China and South Korea still get really hostile whenever Japan does literally anything today. In the early 90s, that was still more fresh, and likely to have repercussions.
However, lets just isolate that and focus on the military issue. The Russian Army was in a state of collapse almost worse than the state was at this point in time, but in terms of sheer assets and manpower, they dwarfed what Japan could bring to the table at the same time. Japan's ground forces are not equipped for amphibious warfare or really for any kind of foreign deployments. Its naval and air assets are significant enough to cause trouble for any Russian attempts at reinforcement, but Russian missile forces of the non-nuclear kind could still do immense damage to Japanese bases and gradually eliminate those assets.