I study ballet in the Soviet Union, so this is - at last - about the perfect question for me to answer.

Let's start with the 1920s. In the 1920s, ballet was a somewhat hotly contested realm. Some hard line Bolsheviks/Communists did take the position you propose - essentially that ballet was a bourgeois (although I'd argue it's more aristocratic than bourgeois) art form that had no place in a socialist world. According to this position, such art forms really couldn't be made to conform to socialist ideals. On the flip side, there were classically trained artists as well as those who felt that high culture was in fact worth preserving. Anatoly Lunacharsky was the Commissar of Enlightenment (yes, Enlightenment), and was a major voice that spoke up in defense of ballet, opera, etc. Lunacharsky is a relatively well known figure by this point, at least since Sheila Fitzpatrick's dissertation then book The Commissariat of Enlightenment. Inside the actual academic theaters in Petersburg and Moscow there was a lot of discussion over this as well. Many of the artists seemed genuinely willing to try to push their art form in a socialist direction, while others thought that ballet was a robust enough form that it did not need substantial changes to fit into a socialist world. In either case, in the 1920s classical/romantic ballets continued to be preformed. The Communists went out of their way to try and reach working class audiences. Communism was meant to be rule by the workers - but many, Stalin included in turns out, felt strongly that those workers should be well, "cultured" (they used the term kul'turnost' in Russian). Workers did go, but it was often less popular than other art forms that had a more popular character (like film) - for a discussion of popular film see Denise Youngblood's Movies for the Masses.

New ballets were also created, albeit at a relatively slower pace, with the goal of finding a socialist version of ballet. Meanwhile, at the ballet schools themselves, classical ballet technique continued to be taught, which meant that the new generation of ballet dancers were being technically trained in more or less the same fashion as the generations of artists before them. Keep in mind that, globally, ballet was far from stagnant even before the revolution. The Rite of Spring came in 1913 and was considered extremely radical. So in some ways, the early socialist ballet was even more conservative than forms of ballet being practiced elsewhere. Diaghilev's Ballets Russes might be seen as one direction of Russian ballet (that strongly influenced European and American ballet in the 20th century), and the Soviet ballet as another.

Socialist Realism wasn't made official until 1934, so there was quite a long time when there wasn't a clear state mandated standard for form. In fact, socialist realism was first articulated in terms of literature, but the aesthetics of socialist realism were often not clear either way. Ballet underwent a period where dramatization was emphasized at the expensive of "dancier" elements, but this was a temporary stop along the way. It's important to realize that formalism in all of the arts was decried by this time - the effect of which was that often times well established easily approachable forms continued on. Abstract art, which had enjoyed a brief period of experimentation in the 1920s was demonized as bourgeois decadence, paving the way for those well-known 19th Century realist forms to become acceptable under "socialist realism." However, it should also be said that the standards for what was an was not acceptable were not often clearly articulated either way. There was more "wiggle room" than one might expect. For example, a colleague of mine's (currently in progress) dissertation goes into a great deal of detail about the way artists and censors went back and forth to try and find acceptable forms in opera. Artists were not simply thrown in the gulag for getting it 'wrong' - they often spent years revising and revising until they hit on something. This way of operating continued until the early 50s. Socialist Realism was in practice much more a process than easily defined standard.

After Stalin's death things changed fairly dramatically, and although you do mention the Cold War once, the rest of your question feels very much like you are asking about life under Lenin and Stalin. The ballet and opera continued to be practiced during the Cold War, but much of the hand wringing over the kinds of questions you are asking had ended by the time Khrushchev took over. By this time questions of culture often took on a more geopolitical nature. In other words the Soviet Union wanted to prove that it could produce just as impressive culture as anyone else, so these classical forms could be help up in that context as evidence that the Soviet Union was not some "culturally backward" country. (There is a ton to say about what really constitutes "cultural backwardedness" and whether or not such a thing actually exists, but suffice to say that I think its largely an incoherent idea) .