The political factors would determine the outcome- Do the smaller German states still back Prussia? The defection of the South German states would make Prussia's position very dangerous
They are committed to Prussia by the treaties of alliance signed in 1867. Also, German nationalism is growing steadily in ferocity and any King who took the French side would risk losing his throne. Many Germans would prefer Austria to Prussia, but a choice is between
Germany and France is no choice at all.
Do Austria and Russia stay neutral?
Probably. If Russia comes in it will be on the Prussian side. They have no reason to support France, and on this TL Prussia has probably taken their side in 1878, since it needs to keep them onside until unification is complete. Austria may be tempted, but that would risk bringing Russia in against her. Franz Josef has learnt caution.
Using Otl as a guide, the French were often stronger than Prussia in the period 1870-1914. If the French had spent a little extra money and adopted better strategy, they would have easily beaten the Germans in 1914. We can be certain that France would be much stronger against Prussia in 1886 in the TTL than in OTl if for no other reason than not having paid the indemnity
OTOH Germany's population has increased abt 15% since 1870, while France's has hardly changed. So the disparity in manpower is growing.
Also, even if France does manage to win, what can she do with the victory? By attacking Germany, she has whipped up a storm of hatred which will make unification inevitable if it hasn't already happened. And even if she annexes a chunk of the Rhineland, that won't change the balance of power enough to matter. Expect a German war of revenge before long, with public opinion in most of Europe favouring the German side.