Darwinism occupied a tricky position in Stalinist USSR. On the one hand, Stalin was himself a self-proclaimed Darwinist, and the Soviets did hold that natural selection and materialistic evolution in general was the correct description of biological evolution. However the Soviet philosophy of dialectical materialism, as implemented by Stalin, basically said that no natural science could impinge upon the domain of the state, which is to say the "science" of Marxist-Leninism and its economic, historical, and political representations of the domain of mankind. Which is to say, social Darwinism would have been completely unacceptable, as would be eugenics, or any other place where you might try to make the move to say, "because nature tells us X about humans as creatures, we should organize them in a different way than the Marxist-Leninists might tell us to." Darwin's own science was seen as somewhat bourgeois (because Darwin was definitely bourgeois by Marxist standards), and he borrowed (in their eyes) too much from Malthus and others that were considered to be politically suspect.

Now I might wonder if you (or Radiolab) got the "open Darwinian" thing wrong — was he an open Mendelian? Because that is a different matter; Mendelian genetics was essentially banned under Stalinism (for a variety of political and ideological reasons), and it could make for a very difficult time for anyone who practiced it openly.

On dialectical materialism, and Lysenkoism, see Loren Graham, Science in Russia and the Soviet Union: A short history (New York: Cambridge University Press, 1993). There is also a chapter on how Russian intellectuals interpreted Darwinism, but it is pre-Soviet.