I think we'd need a different Emperor. Never gonna happen with Franz Josef. Maybe he dies young and the ill fated Max ends up Emperor of Austria rather than Mexico. He was quite liberal; indeed, he was removed from his post as Viceroy in Lombardy-Veneto because he was pursuing too liberal policies for the Emperor and his cabinet.
He also loved the navy and Trieste.
He may be just the man to change things.
The only problem with that is that you'd need the succession to go through Ferdinand, Franz Karl, Franz Joseph, and then finally end with Maximilian.
Perhaps if Franz Joseph's assassination attempt in 1853 was successful? I can't see Maximilian being lured into the
Campagne d'Italie the same way Franz Joseph was, and given his liberal policies in Hapsburg Italy the moderate and conservative Italians might be swayed away from the
risorgimento. Milan is likely still going to change hands, I don't see the epicenter of the movement staying under the Hapsburgs sway, but Venice and Tuscany might stay within the empire.
The problem though is that this doesn't result in a Hapsburg monarchy with three crowns as the OP asks for, nor the federal commonwealth I was speaking of. At the least you'd need the initial POD, then for Maximilian to do very well and overcome the reactionaries in Vienna (and Zagreb) and make some moderate liberal reforms, and even then you'd need him to be followed up by an heir who was just as much a liberal reformist. And all the time the Hapsburgs would need to not get bogged down in foreign entanglements, while keeping their restive populations content, and to keep up, both in power and prestige, with the other great powers. That's a fairly tall order.
EDIT: Maximilian's Wikipedia page indicates that Maximilian & Charlotte's adoption of Augustine I's grandsons was actually a gamble intended to force his cousin, Karl Ludwig, to give up one of his children to the imperial family as an heir. Perhaps ITTL Maximilian, who is now the Hapsburg emperor, simply decrees one of Karl Ludwig's children to be his heir - which likely means Franz Ferdinand.