Prussia Stand-In Wanted

One thing to keep in mind is why Austria and Prussia were able to grow to be such powerhouses in Germany: they both had territories outside the HRE that were not influenced by HRE politics and that they could rely on for manpower and support. So to have a third (or in this case in Prussia's absence, a second) big German state, you need one that can lean on non-HRE territories to draw some power from that they can swing around. Some suggestions:
- Sweden that rules key German territories, such as Brandenburg
- A state that rules the Netherlands as well as territories in the Rhineland (Hanover?)
- Bavaria that holds some of Northern Italy
So did the rulers of Saxony and Hannover.
It did not make much of a difference.

Austria and Prussia surpassed others thanks to a combination of a fairly effective expansion, consolidation & influence policy within the Empire.
An example of how latter's expansion within the Empire contributed to its increasing influence.
The Hohenzollerns instead kept their possessions distinct to maximize representation both in the Reichstag and in the Westphalian and Upper and Lower Saxony Kreis Associations.
Formal political influence was important, because the Hohenzollerns could not compete with Habsburg informal patronage. Although large, the Hohenzollern army and court lacked the prestige and the number of well-paid appointments the Habsburgs could offer, while the emperor had much wider powers of ennoblement.
Representation in the Reichstag provided a platform to rally support to block uncongenial Habsburg measures, as well as to legitimate Hohenzollerns policy on a wider stage.
- Peter H. Wilson's Holy Roman Empire, p. 474

Their territories outside the Empire (Hungary and Ducal Prussia) helped, but as extensions of their German realms.
Well, in Antony444’s timeline “In this country, it is good to kill an admiral from time to time”, Saxony becomes the Prussia stand-in since Prussia was dissolved after a war in 1765. They become by far the strongest of the 4 German states (though tbf 2 of them, Westphalia and Dutch Germany, are French puppets and the last one is Bavaria, who was the strongest state until it got defeated in the 1840s and lost a lot of land).

During the Great War, they work with Austria Hungary to invade the rest of Germany. They easily overrun Bavaria and initially overrun most of Westphalia, but the French juggernaut stops them at Ulm. After 4 horrific years of war, Saxony is beaten, and now Westphalia is the strongest of the 3 German states due to being a French puppet.
Assuming that had actually happened, what other OTL state, Austria aside, would have had the best odds of managing to fit in Prussia's shoes?
Depends on what this means.
Dominant power in north-eastern Germany which expands at the expense of Swedes and Poles: Wettin Saxony
Main rivals of the Habsburg in Germany: House Wittelsbach

Both were attempted IOTL, but were not sufficiently successful to achieve lasting success, unlike the Prussian Hohenzollerns.
It was all about "Joseph wanted more lands for Austria". Maria Theresia didn't want to weaken fellow Catholic state in favor of Orthodox Russia and Protesstant Prussia.
Austria annexed Szepes County in 1769-70, we’ll before the 1st Partition. If anything, it was CII who was reluctant about the whole issue because it was to the Russian disadvantage: the whole PLC was Russian vassal. MT and Kaunitz are always getting “credit” for the 1st Partition on the Austrian side and, anyway MT was Archduchess of Austria and looked for the Austrian interests. Initial initiative was coming from Prussia but MT jumped onto the board before CII and joined Austria-Prussian pressure forced her to agree.
I don't think you can have the Commonwealth be a superpower under anything like its OTL system of government.
Yes, it was in a deep trouble well before Brandenburg-Prussia raised into prominence and successes on the East only helped to ignore the systematic problems within the PLC. Strictly speaking, while the PLC managed to get few pieces of territory from Tsardom during the ToT, strategic value of these territories was quite limited and the main goal (Putting Wladislav on the Russian throne) was not achieved. The Cossacks “issue” was not going anywhere (even during the ToT Sagaidachni tended to act more as an semi-independent ally rather than as an obedient subject), Livonia was lost to Sweden (with a resulting control of the Lithuanian exports) and it was just a matter of time for Tsardom to recuperate enough to start taking back the lost territories.

Even a short-term raise to importance under Sobieski was a fluke: he needed Hapsburg subsidies to keep his army together during March to Vienna and after Vienna the Polish-Ottoman hostilities were not going anywhere, as I understand, due to the lack of funds and general resistance to a perspective of having a strong ruler. By the end PLC got Podolia (IIRC, even the Ottomans did not manage to squeeze any profit from that region) and lost Kiev forever. And look at the mess that followed....

Prussia still was pretty much irrelevant.