Is it just me or did he lose the Hamlet thread? It doesn't seem so much a mystery as a purely strategic post-war moment that no really knows about.
It seems around 1940 or so Stalin did stop a Hamlet production (based on a Pasternak translation) at the Moscow Art Theatre. Recollections vary somewhat but the most plausible explanation is that Stalin felt Hamlet (the character) was weak. Completing the circle, the Russian wikipedia page on Ivan The Terrible has a quote from a Central Committee critique of a different movie in which, as an aside, part two of Eisenstein's film is criticized as ahistorical and as portraying Ivan as too weak "like Hamlet". The second part was not screened until some years after Stalin and Eisenstein's deaths. Hamlet itself was not staged in the last 12 years of Stalin's life.