Wouldn't that scenario lead to a war between Prussia and Austria anyway? Austria was NOT going to accept a Kleindeutschland easily, and WAS very close to going to war with Prussia in the early 1850 over a much more watered down variant of it. Likewise, a lot of (Klein)Germans would se Austria as unfinished business to be fixed ASAP. .
I don´t say a german/austrian war can´t happen, but I wouldn´t say he is inevitable. Even IOTL Austria had to wait till 1850 before it feel strong enough to search the conflict with Prussia. At this time the prussian Unity-politic had already failed and Austria just had to set the deathblow. But ITTL the new German Empire would be already established. Breaking it apart would be difficult ans risky for Austria. Russian help would be needed in any case and even if it was successfull it could mean that Austria would become a russian vasall.
In such a war ca. 1850, Austria would be likely supported by Russia, but it's hard to see what horse France and Britain would have in that race.
Surprisingly Louis Bonarpard offert IOTL 1850 Prussia an alliance against Austria and Russia. Being himself he wanted to get the bavarian Palatinate for this, but I wouldn´t rule out that he would like to be the Defender of peoples rights against the powers of autocracy. Britain had not much symphaty for the german unity movement (especialy in the Schleswig_Holstein question, but if wouldn´t also like the idea that Russia establish a full Hegemonie about Central Europe-