The Franco-German War (I don't like the term 'Franco-Prussian War') is one of the most one-sided wars in history. To reverse its outcome with such a late PoD is really hard, for several reasons.
The Royal Prussian Army was at the time the best in the world. Prussia had universal conscription, a thoroughly well-organised system of army reserves and excellent infrastructure, which allowed it to raise huge armies at short notice and deliver them exactly where they needed to be. Prussia also had a general staff headed by Moltke the Elder, widely considered one of the greatest military geniuses of all time, who gave the Prussians (and thus the Germans) planning ability that France was utterly unable to match; lots and lots of what is now considered crucial for a modern general staff is owed to Moltke.
As for foreign intervention, I don't think it's likely. Austria was tied up in the business of the Compromise and the Austrians knew full well (they'd just had it proven to them!) that they were absolutely no match for Prussia. Denmark might intervene, but that's not particularly relevant. Russia and Britain considered it no concern of theirs until the Germans absolutely trounced France, which ruined the post-Crimean War consensus of Anglo-French alliance against Russian expansion.
For a PoD after the beginning of the war that permits France to win, you would need some extraordinary bad luck for the Germans—perhaps several important people (let's say, Bismarck, Moltke and King Wilhelm I of Prussia) are all killed, by freak chance, by French artillery near the beginning of an important battle.
I do think there are PoDs earlier than the beginning of the war that could quite plausibly give you a French victory, and I'm planning a TL on a similar basis. But I can't think of any reasonably likely possibilities for this challenge as it stands, and I'm curious to see what Anaxagoras had in mind when he said it wasn't impossible.