Most of what you say makes sense, but it would take more than a single defeat to force Prussia accept such a great reduction, when both Britain and Russia strongly disagree. Russia can be theoretically bought with concessions about that straits, but that would make Britain even more furious. I concur that both Austria and France would have great convenience in a Carthaginian peace, but I am not sure they would consider it worth the necessary war effort, that won't be only a reverse Sadova. If Italy is still at war, it would be even more costly, and in your proposal would put Napoleon in very weird position, advocating the gift of his ally's territory to someone he himself is at war with. And Garibaldi may actually decide do some weird stuff in Nizza (his hometown) or more likely in Rome, if France acts against Prussia.
France doesn't have to declare war on Italy. Indeed, it could even occupy the Rhineland without declaring war, though it would be unusual behaviour.
Even if Prussia did try to fight on what would she use? Her best troops are in Bohemia and can extricate themselves only through narrow passes, which will take time if it's possible at all. More likely they are pinned against the Bohemian mountains and forced to surrender.
Iirc the troops in Western Germany are mostly Landwehr, which performed indifferently even against Hanover and Bavaria. So the French army can probably reach the Elbe before running into real opposition. Bismarck himself, in his memoirs, stated that French intervention (even after
victory at Sadova) would have compelled withdrawal from Bohemia in order to cover Berlin.
Afaics, if Prussia loses Sadova, she is toast. That is probably what Von Moltke meant when he told his king (who was getting jittery and talking about retreat) that "We are fighting here for the very existence of Prussia".
As to foreign powers, the crucial thing is that they all had other fish to fry, and that German affairs were way down their priority list. Nappy III's attention, as noted, was focused on Italy. The Tsar was mostly concerned with shaking off the Peace of Paris, and would also have neen terrified of a French army on his Polish border, so soon after the revolt there. As for Britain, given her troubles with the Union in the recent ACW, and the current Fenian raids into Canada, she is far more worried about possible war with the US than with anything happening in Europe, esp Central Europe where she has little or nothing at stake. Italy, of course, has been stomped at Custozza, and with her Prussian ally also beaten can only beg Nappy III to intercede for her. In short, the only country Franz Josef has to worry about is France, whose acquiescence can be bought by yielding Venice.
Could I recommend AJP Taylor's
The Struggle For Mastery In Europe, the relevant chapters of which cover the diplomatic ins and outs pretty well? Geoffrey Wawro's
The Austro-Prussian War is the best account of the war itself.