Let's hand-wave the following the situation. Louis-Napoleon simply is not involved in the Mexican intervention scheme in the 1860s, so he is more sensitive to events on the continent.
Not liking Prussian assertiveness, and seeking profit for France in the Rhine territories and ultimate freedom of action in Luxembourg, he declares war on Prussia when Prussia declares war on Denmark. Since Austria (and the rest of the German confederation) is allied with Prussia at this time, Louis Napoleon, entices Italy into the war by promising them Venetia for going to war against Austria to tie down their forces.
How does this war go militarily? What is the longest it can last? What attitude will Britain and Russia take? They were broadly pro-Danish by inclination, but couldn't justify supporting the Danes in this particular instance and the Russo-Prussian relations were excellent at the time. Russia disliked both France and Austria at the time.
Not liking Prussian assertiveness, and seeking profit for France in the Rhine territories and ultimate freedom of action in Luxembourg, he declares war on Prussia when Prussia declares war on Denmark. Since Austria (and the rest of the German confederation) is allied with Prussia at this time, Louis Napoleon, entices Italy into the war by promising them Venetia for going to war against Austria to tie down their forces.
How does this war go militarily? What is the longest it can last? What attitude will Britain and Russia take? They were broadly pro-Danish by inclination, but couldn't justify supporting the Danes in this particular instance and the Russo-Prussian relations were excellent at the time. Russia disliked both France and Austria at the time.