I mean at least later on they did try to ally with the UK. Maybe that was the long term plan and they didn't expect things to go the way it did with the naval war. Don't know, good question. But the French allied with Russia of course and they got away with it and got UK support.Fair enough. That said, though, if the goal is merely stability, why not ally with Britain instead of with Russia? In fact, this might have actually happened had WWI been delayed long enough or not happened at all.
Germany would obviously require the Hapsburgs in Austria to renounce their rights to the Hungarian throne. A Hapsburg cadet branch could have took over Hungary in this scenario and renounced their rights to the Austrian throne in the process. The Hapsburgs had a lot of males and thus having some of them renounce the succession in some of their realms would not have been a huge deal for them:
Line of succession to the former Austro-Hungarian throne - Wikipedia
Also, how much people did German Austria have relative to Bavaria? I mean, if Bavaria had more people, then it rather than Austria could have been the leader of the Catholics in a Greater Germany.
I don't think the Habsburgs would be willing to do that to become a glorified fief to their former glorified fief(though I guess some would argue within the Central Powers they became that to some extent). Just didn't make sense for anyone involved.
Bavaria today has more people(about 20-25% more doing guess math in head) but is slightly smaller. Didn't check for 1914 but the difference in power doesn't come from Austria itself being larger and more important but from the collection of territory the Austrian King had outside of Austria. And Bavaria came closer than any third German state to doing that but never happened. The Bavarians tried to become the rulers of Germany during the War of Austrian Succession and were supported by the French so there's a universe where this could have occurred but it didn't work out for them obviously while Frederick's jockeying for power in the same war did work out. Interesting to think about but the door to Bavarian domination of Germany had shut sometime in the 1700s or at the very latest when Napoleon was defeated.