Not likely, I'd even say impossible. And it's not only because of state censure. It of course would be a serious obstacle if some "Soviet Beatles" actually appeared, but I doubt that anything would have appeared at all.
There were a lot of various soviet popolar music performers which enjoyed a huge popularity in USSR and not all were directly governed by the State. But none AFAIK piqued any significant interest abroad. The problem was (and is), I think, in cultural background difference. There is a part of russian and east european cultural tradition which is not shared with western cultural tradition. On the other hand, western culture had huge impact in Russia, but still have not become a true native background.
Which means that while western performers may be very popular in USSR (and often were, but, interestingly, quite often not the most popular ones in Europe/US), they still would not be as popular as some local ones. And soviet performer would be seen in the West as either not very original, or too exotic. It is concievable that some genuis could fuse both culture layers and rise to fame in both USSR and West, but he still would lack true mass popularity.
The point is that while there may be "Soviet Beatles" to shake the USSR, it is very unlikely they would succeed in Europe/US in the same manner. And the performer, who could succeed in the West would have a hard time impressing soviet fans.