I wasn't joking about Austria integrating Prussia. It is possible. Until 1806 Vienna was even Prussian's capital (Kaiserstadt Wien). The Vienna population is even nowadays joking about, when they meet Berlins, btw.
Prussia did not consisted of one folk, the Germans. In Prussia lived a lot French protestants (Hugenotten), Polishs (in Rhinish provinces, Silesia, Westprussia, Warszawa), Silesians, Sorbishs, Lithunians, White Ruthenians, Swedishs, Danishs, Plattduitsche (Friesland) etc. Until 1918 every Prussian law was adressed at 'The folks of Prussia within Prussia'. But even these Minorities accepted Germany as a State, because they knew, that their rights were heavily protected within Prussia and that the federal State had now possibility to intervence within Prussia itself.
In my mind and even, if Prussia lost the war against Austria, there is a need for a possibility for Austria to integrate Prussia into a German state. Do you know, why the OTL and even your TL called 'Niebelungentreue' loyality between Austria and (Prussian) Germany existed? Because Austria AND Prussia and their inhabitants recognized themself as the two German States. And if they did, they are able to unite under special circumstances.
IMO if Prussia or Austria unite all German lands some kind of forced 'Ausgleich', renogitiation, between the non-German folk and the Germans should happen (if fail=civil war). Only Bohemia, Polish territory and Slovenia were integral parts of Older Greater German area. Hungarians & Slovaks, Romainions, Galicians and Ukrainians and Russians would revolt against Greater Germany or negotiate for puppet-states.
At the LP-Prussia-issue: Prussia became a great power because it waited, used tactics and never got involved into 'Volkskriege' great Nation-Wars. So Prussia SHOULD have the chance to ally with LP and sit it out.
Aaaaand... your TL is to cool to not have an aftermatch with inter-war-period and WW2^^