The three main Shogunates in Japanese history were based at Kamakura in the east, Kyoto in roughly the center of Japan, and Edo in the east. What would be the easiest and most plausible way for a shogunate to be established in the west of Japan? I'm not sure what the definition of Western Japan technically is, so let's use the area west of Lake Biwa as a starting definition. I think Kyoto might be considered western Japan, so let's go with something that's as west as possible. A bakufu based from Shikoku or Kyushu would be best.
The period after the Kamakura Shogunate saw a lot of powerful clans in the East, so I think that period is out. Perhaps a different Onin War sees one of the families in the west take power from the Ashikaga?
And no, I don't mean a Westernized Shogunate. Please don't say something about modernization. If that is the end result of a hypothetical scenario, that's fine, but I'd actually like to see an earlier point of divergence.