Not much more to add, unfortunately. Only thing I can remember about this story is that is was told from a Soviet POV about some scientific endeavor in genetics, possibly Lysenkoism-based. The "speculative history" part of this was that Lamarckian genetics was correct, not Mendelian. The Americans still believed in Mendelian genetics/survival of the fittest, which got into a nasty feedback loop based on the fact that they reproduced Lamarckianly too. I think it ended with an American supersoldier/spy breaking into the facility?
Could this be "Red Legacy" by Eneasz Brodski? Published in Asimov's Science Fiction in 2015, it deals with an alternate-reality Cold War setting, where the Soviets indeed find that Lamarckian genetics is correct. The science base where the cloning experiments were conducted, the Arkhipov facility in the Urals, was subjected to attack from British spies (MI6) and American commando forces.
The story is also available in a Kindle edition which has a few pages as a free sample, which may jog your memory.
It's not as good a match, but another Soviet-biologists-prove-Lysenkoism story is "The Lysenko Maze" (1953) by Donald A. Wollheim (writing as David Grinnell).
The Soviet researchers, in a farmhouse outside of the university town where they normally do research, construct a very complex maze in which they will breed mice. The maze is complex, resources will be moved around following complex patterns, there will be not quite enough food, or heat, so that the mice will be forced to be smarter to survive.
Ultimately they manage to breed mice smart enough to understand that humans are controlling their environment, what electricity can do and how to defeat the ultimate barriers (physical, chemical, electrical) that imprison them. In the end the researchers burn down the building in an unsuccessful attempt to keep them from escaping.
It has been anthologized several times in fairly diverse collections, and it's possible the super-soldier is from another story in the same collection. You can read the story in Amazing Stories, July 1954.