The plausibility of this depends on whether you actually mean 'stay a monarchy' or just 'be a monarchy'. If you mean that France has to stay under the Bonapartes, I doubt it; that would require French victory in the Franco-Prussian War or the avoidance of that war altogether, and in either of those scenarios there wouldn't be a Germany as we understand it. If you mean that France merely has to have a monarchy, that's easy. Kill off the Comte de Chambord after the Legitimist-Orléanist agreement (perhaps to stay within the boundaries of this scenario he can die of a disease that he catches in 1869), and you can have King Louis-Philippe of an Orléanist monarchy.
As for a German republic, this depends on SvoHljott's question. If it's just the PoD that has to be in that time, the easy solution would be to have history go roughly like OTL, leading to the collapse of the German monarchy and the birth of the Weimar Republic. But if there has to be a German republic by 1870, I have no idea how to accomplish that.
Alternatively, we could try to do something wacky with the revolutions of 1848 succeeding somehow, but I prefer the dead-Comte-de-Chambord scenario above.
{edit} I don't think we can change the personal nature of Napoleon III with a PoD after he was born and raised. And as for a scandal, I don't know whether it would have been enough to topple the German monarchies, but Wilhelm I might help you there: he was a barely-controllable eccentric whom Bismarck had to keep on a short leash, more belligerent than his famous grandson in temperament, and he genuinely wanted to annex the Habsburg empire once he'd won the Austro-Prussian War (Bismarck managed to stop him), so he's exactly the sort of personality who might create a royal scandal.