The Austrian behavior during the war of Crimea was anything but linear: there were different interests at play, a general lack of money and overall a strong urge to secure the borders before committing anyway.
The potential dangers to Austrian borders were Prussia and Sardinia. Theoretically Prussia was an ally, at least in German matters; however the humiliation they had been subjected to at Olmutz (being forced to scrap the Union of Erfurt and accept Austrian presidency of the German Confederation) had not been easy to swallow. Sardinia was an obvious enemy, even if the rebuilding and reforming of their army was still under way; while it could not face Austria alone, it might take the opportunity of an Austrian involvement in the war against Russia to strike, possibly in conjunction with insurrections in Northern Italy.
The Austrian Foreign minister, Buoi, was in favor of supporting France and Great Britain in their war but had met the firm opposition of the army who did not consider feasible to commit troops in a major war. There was also the problem that the Russo-Austrian alliance had been a pillar of European diplomacy since the Congress of Vienna. OTOH supporting Russia, even leaving aside the money problem, would not have furthered in any way the Austrian strategic interests (navigation on the Danube and influence in the Balkans). Buoi had worked secretly to obviate these issues, and by the end of 1854 an agreement was signed with Prussia, while France and Great Britain were working on Sardinia (who had been convinced to join the western alliance). Prussia had been the easiest problem to solve (although Prussia did not promise much), while discussions with Sardinia took more time. Anyway Austria mobilized at the end of November, and signed an alliance with France and UK by Christmas 1854, promising an Austrian intervention unless Russia accepted peace on the basis of the Four Points agreed with the allies (Danubian Principalities to be guaranteed by European Powers, free navigation on the Danube, revision of the naval agreement of 1841 and Russian renounce to any right to protect Christians in the Ottoman empire). Sardinia signed a military convention in January 1855 (there was not a lot of other options for them, since allowing Austria and France to become closer would have been a disaster) and diplomats met in Vienna in February to discuss the peace proposal based on the Four Points. Strangely enough the 1st and 2nd points (which were the ones most sought after by Austria) were not much of an issue: the front of the Danube had collapsed, and the Russians had been forced to evacuate the Principalities. The 4th point was also not much of a problem, but the 3rd one was much stickier: after destroying the Turkish fleet at Sinope, the Russians were dominant in the Black sea (excepting only the presence of Franco-British ships). The discussions on this point dragged on until early June without any result, and the Russians left Vienna on the 10th. At this point Austria should have honored the commitment of the December Alliance and moved against Russia, but surprisingly demobilized. It is quite obvious to me that two years had not been enough to put the Austrian army in condition to campaign against Russia and the November mobilization had been a bluff, but it is also equally obvious that the Austrian turn-about was not well received by the Franco-British: in just 6 months the Austrians had managed to damage their relations with both the parties involved in the war. No need to go into details about the rest of the war: Sevastopol siege finally succeeded, and the city fell. In January 1856 the Powers and Sardinia met in Paris to negotiate the peace treaty (which was signed on the basis of the Four Points, including an almost complete neutralization of the Black sea). From Sardinia POV, it was possible for Cavour to sit at the victors' table, and to raise for the first time the Italian issue. From the Austrian POV, they got what they wanted on the Danube and in the Principalities, but substantially damaged the relations with Russia and were unable to get closer to France. It may look like I'm in the game of Buoi's bashing, but in all honesty he was dealt a very bad hand and his moves were more or less forced (which also in my opinion makes any kind of POD very unlikely: what could have he done? The alternatives to the bluff were only intervening in June 1855, but it might have ended very badly for Austria - maybe even something like a pre-view of Conrad's offensives in the east).
Archduchess Sophia now: from what I gather (sources are a bit scarce on the ground) she was a cold and ambitious woman, as well as a conservative one (it is alleged that her coldness grew a lot after the death of the king of Rome - and the legend that FJ would be the outcome of a dalliance with him was never proven and must be considered a legend). In 1848 she managed to convince her husband, archduke Franz Karl, to renounce the throne in favor of their son FJ (Franz Karl was apparently not the sharpest tool on the rack and was not much interested in politics): afterward she was the power behind the throne, and it is said that FJ defied her only once (when he insisted in marrying Elizabeth, rather than her sister Helen who was the one his mother had chosen). Her hate of Hungarians was very well known. It is said that Maximilian was her favored son, and it is quite possible. OTOH when Maximilian proposed reforms in Lombardy-Venetia she certainly did not goto bat for him. Archduchess Sophia is another very good reason why it is very unlikely that Maximilian will be able to implement significant reforms in the empire.
Sorry if I cannot give you good news: I understand you're trying to find a way of reforming the Austrian empire, but I'm afraid that the deck is a bit stacked against you
