the bourgeoisie or the proletariat.

There's more than just those two options, e.g. the nobility (which we don't have anymore). If you had done some research on the topic, you'd know that a class is a group that shares a common relationship to the means of production. You could perhaps argue that career politicians constitute a sort of "political class".

The USSR had democracy - there were voting structures up and down the entire structure of the union.

In their worst election (1946), the Bloc of Communists and Non-Partisans received 99.19% of the vote. I don't know who their opponents were, the other 0.81% of the vote simply went to "against". Results like this are impossible in a free and fair election.

The communist party of the soviet union was in control of the country, with absolutely no accountability to the proletariat they were supposed to be representing. Were they working class? No, because they did not work. They were a new kind of nobility, with the head of state effectively being a new Tsar.