I think you would need a Japan that was more concerned about domination from its neighbours, so that it would accept more influence from western nations in order to stave it off.
A Chinese Empire with a more muscular and imposing foreign policy, that sought to vigorously enforce the tributary system ( and with the political will and military might to match) in the face of western encroachment might be the best way to produce this.
Even then barring major social upheaval such as a second sengoku, or a sectarian civil war between Christians and Shintoists, I don't see the imposition of direct colonial rule as a likelihood. Rather that Japan would become akin to a princely state writ large, as it becomes economically and militarily dependant on Britain.
This state of affairs is unlikely to last long as I think Japan would begin to industrialise despite this, though perhaps slower. Once such efforts came to fruition though, Japan would inevitably drift away from the Empire. By the 1920s it would almost certainly be basically equivalent to a dominion. The OTL japonic craze, and respect for the Japanese as 'almost white' will probably aid this process.
The resulting Japanese state would however have much stronger economic and political ties to Britain.