Why would the Dutch stick with the Russians? Their vulnerable trade routes would be liberally pillaged by British ships.
Also I don't think the French would ever be in such dire straits to consider that option.
The main revenue-bringing routes of NEI (Netherlands East Indies) were local east Asian routes. The disruption by of Dutch trade routes by British may be not fatal if Dutch manage to control the choke point of Singapore. The role of French in Japanese rise was important, but not critical. Anyway American Civil War ends in 1865, and US-British relations in 1865-1870 were "chilly". In ATL, US can take the place of France if needed (which i doubt, because of the politics of French IOTL in Crimean War of 1855 was more self-centred rather then pro-British)
IOTL, Dutch had a long history of concessions to British, because direct opposition would be suicidal. But Britain weakened by indecisive outcome of Crimean War of 1855 (which was a war won by force multiplier (therefore susceptible to technological invention-dependent POD)) may be perceived as a more vulnerable. And vulnerability attract the scavengers.. in this case the Dutch-Russian-Japanese alliance as i propose.
Also, in proposed ATL, the Russo-Turkish War (1877–78) is going to turn out a catastrophe for Ottomans (may be up to occupation of Istanbul), resulting in Russian control of Sea of Marmara and setting the ground for the anti-British coalition intervention in the First Boer War in 1880.
My basic strategy to find solution for this challenge:
0) Prevent too-early POD resulting in great changes (i.e. it is too easy to develop a timeline where British were never a dominant world power)
1) Find the latest POD reducing the economic power of British to the point it can be confronted.
2) Assemble a coalition for actual confrontation (including Japan as requested)
3) Ensure eventual domination of coalition by Japan by including other members with power on decline.
Of course, i understand the challenge is a stretch. From the POD, because the technological and institutional modernization in 19th century Russian Empire was crippled by many deeply entrenched factors. The most important was the serfdom system not broken until 1861 (end even in 1861 the serfdom end was grossly unpopular among Russian elites). Dutch, on the other side, had a terrible problems with corruption and legacy financial systems (see
https://en.wikipedia.org/wiki/Economic_history_of_the_Netherlands_(1500–1815)#Final_crisis ) Even Japan appears to be less handicapped during the era.