What would a Franco-Russian war in the late 1800s look like?
Well with the scenario I outlined most of the conflict would be in the Pacific, so France's superior navy trounces Russia's. On the other hand though they're a much longer way from home with limited supplies, and if the Russians can get a large force on the island Ezo is lost. Then there's also the Japanese, who are likely going to be allied with Russia ITTL. In a short war (4-8 weeks) I'd say France wins, but anything else is a Russian victory, all others things being equal.
Perhaps in a long war you see French naval operations in the Baltic, but that'd require both Britain and Germany/Prussia being agreeable to the situation, which is... unlikely. By the same token though neither of them is likely to want the Russians sallying forth from St. Petersburg into the North Sea so the war is likely going to be purely a colonial one,
if no other major powers get involved.