I'm assuming that in that scenario it works on the basis that each state retains command of its own army , but with a "If Germany is in danger everybody has to pitch in" deal . Under that scenario, Austria could use its own army, independently in Hungary, or the Balkans (for example). Or, try to get some or all of the other German states to pitch in .
If there is a unified army with a single permanent commander, then that is really a unified German state. I can't see that happening with Austria in charge unless Prussia was well and truly knocked out. Some sort of massive defeat, and probably territory stripping.
What would hold Prussia back, would be that if FJ gets Hungary well and truly brought into line, that makes him (FJ) more powerful. Which makes Wilhelm, in comparison, less powerful. He won't want that . Bavaria, same problem. Smaller states probably don't care. I agree that the most convincing argument would be "We Kings need to stick together against these Republican rabble rousers. All for one and one for all" argument. That would be even more convincing to the smaller states.
So on that basis could there be some sort of agreement like What Metternich tried to make the "concert of Europe" at the Congress of Verona into, a sort of mutual defence pact for Kings against revolutions ? It worked for Spain in 1823 and then sort of lost commitment. Such a "Concert" could include other states, which solves the Hungary issue, and would probably tend to keep Germany balkanised, by more or less guaranteeing the smaller states. Arguably, the best time for that would be some time in the 1850s, after the scares of 1848. However, if Sadowa happens as OTL, (nothing in a *"Concert" to prevent it ) then Prussia goes on as OTL. What if Russia not just guaranteed the sovereignty of the small German states but mandated their sovereignty. The states in the *"Concert* could not even give up their Koniglich sovereignty willingly. Easily justified , to prevent revolutionaries demanding "voluntary" sacrifices. ISTR something along those lines actually happened at Frankfurt in 1848. ? It would be in his (The Czar's) interest to keep Germany non-unified, and as part of a *"Concert" agreement it would seem legitimate. Then even if the Franco-German war happens in 1870, Russia could say "No unification". Which was the OP challenge
It occurs to me that one thing could impel Austria to make a play for German hegemony (not saying it would be successful, mind) , is if the Hapsburgs actually lost Hungary in 1848 (or whenever). No Hungary, being the chief cheese in Germany seems more attractive. Sort of consolation prize. That wasn't the OP challenge, though , so it's off topic.