British East India Company, in concert with the British Government decide to invade Japan post-the first opium war, after sending a request to open the markets in Japan. When that is refused, after the First Opium War, the British decide to open the markets forcefully - leading to the complete takeover of Kyushu and the complete opening of trade between Japan and the rest of the world via Kyushu, which is handed over to the British East Asian Company - which is transferred Hong Kong as well.
Eventually the East Asian Company is brought into conflict again, perhaps through attacks on British citizens, or reneging on treaty concessions, or naked rage from the Japanese - leading to the East Asian Company, later backed by the British Army - to fight, with Kristian Japanese troops as well as Company soldiers. With the technological advantage, and Imperial support, the Company achieves victory, controlling Kyoto, and later Toyko. The peace treaty leaves the Company intact, which supports the a Kristian with an English Wife as Shogun, bankrolling them. With Kyushu and the East Asian Company backing the Kristian Shogunate in Tokyo, you have your Gaijin Shogunate.
As the Great Game expands, or the potential for WW1 still taking place, the Kristian Shogunate becomes a major factor that supports the White Army during the Russian Civil War, leading the majority of the efforts of the Siberian Intervention - using the industrialised Japan as the source for upgrading the Siberian Railway to double track. Strengthening the Whites may well lead to preventing the rise of the Soviet Union.