it was Austrians who fought to the bitter end.
Austria was defeated in the same war as Prussia and was humiliated as well. Battle of Austerlitz, anyone? If you butterfly away Austria's defeat, Napoleon's expansion is checked at Germany, thus removing the impetus to do anything to the HRE other than let it stagnate.
If Austria refused to withdraw from the war, they would've been even further humiliated than they were OTL.
Call themselves as "The Empire" on all formal talking within the German and Italian sphere, make a (false) impression to the masses that HRE still exist.
But in realpolitik, that would fool nobody. The German states don't want to be subservient to the Habsburgs, hence why most of them lotted in with Berlin when the time came for the call to unification.
And the other European powers are only more than happy to oblige the reality, and work against Austrian interests in Germany.
- Promote freedom of religion and do not enforce German on Slavs, Italians and Hungarians. Make it clear there would be no enforcement of Catholicism on Northern German.
Austria's history on religious tolerance in the Holy Roman Empire, and ethnic tolerance in the post-Napoleon era don't support this idea.
I'm considering to move Austrian army into Mexico and made it clear that attack against Mexico would be an attack against Austria, and further, all European Monarchy.
Austria didn't really have enough of a navy or army to project to Latin America in the 1860s. They were already dealing with a bunch of uppity Hungarian nobility and Germany's own little nationalist zeal. You can't butterfly away both unless you stop Napoleon's expansion, which would be a whole separate horde of butterflies.
Also, Britain and the United States are...
rather aggressive in protecting their sphere of influence in the Americas.
I am not quite sure of other European Monarchy's attitude against Imperial Mexico
From what history tells us, nobody except for France and Austria cared about Imperial Mexico. Britain was more than happy to play along with Lincoln and his successor's two-string band; Prussia was busy aggressively pursuing unification, and Russia was gladly selling her only remaining American colony to the United States. The Yanks were on the rise and after the Civil War, Lord help the nation that tried to stop it.
only because of Lincoln that Britain did not come in and fight with South. I don't think that's too probable so it's not implemented.
The UK had nothing to gain and everything to lose from supporting the CSA. And if you remove Lincoln from the equation, the Civil War is a whole separate kettle of fish, and a whole different outcome because I doubt any American politician could've pulled off what Lincoln did.
Even if the UK did go to war with the Union and somehow win, what would it do? It make the United States just like France post-1870 or Germany post-1918 writ large. An entire nation, embittered and angry at a foreign power for taking their land and enforcing stupid occupation rules, and ready to strike back. Within a generation's time, the United States will be armed to the teeth, very angry, and ready to bury entire civilizations in her anger.
If France can't hope to bolster Mexico, how the hell could Austria do so?