I'm having some trouble working out an answer to your question, mainly because there's no one single Big Bang theory. Obviously Georges Lemaître, the Catholic priest you mention, didn't just come up with it out of thin air, and he was working with other discoveries that almost proposed the same thing, but even beyond that, there are at least two different models of how the Big Bang happened. I'm going to ping u/restricteddata in case I get something wrong about that part.

So the first thing to note is that a Soviet scientist, Alexander Friedmann, actually played a very large role in developing the Big Bang theory. (You may also find his names spelled Aleksandr and Fridman, respectively.) Basically, in two papers, published 1922 and 1924, he took Einstein's General Theory of Relativity and expanded on it to show that it was mathematically possible that the universe could be constantly expanding and also stable.

Einstein, however, was unwilling to imagine a universe that was not in a "steady state", neither expanding nor contracting, and he rejected Friedmann's claim. (He later backed down when he saw Friedmann's work). At least for this very early period in the development of the Big Bang theory, you might say it was the Big Bang theory that was a radical idea, and thus, fully in keeping with the revolution's overturning of the old, pre-revolutionary world.

By then, the actual proof that the universe was expanding and that it had been doing so consistently was not far away. In 1924, Edwin Hubble observed that galaxies were all red-shifted in proportion to their distance from Earth, meaning they were moving away from us. Then, in 1927 Lemaître took that fact in the context of Friedmann's ideas and derived some more equations that showed that the universe was in fact expanding, and that it had done so from some point in the past when it was infinitesimally small — a "primeval atom," in his words.

At first, the Soviet reaction to this idea was wholeheartedly approving, in part because it wasn't viewed as some external, Western, bourgeois idea yet, and either way Soviet society was in fact surprisingly ideologically fluid and open in the 1920s. Building off of Friedmann and Lemaître, Soviet physicists lead the field of cosmological physics for nearly the next decade. Matvei Bronshtein and Lev Landau in particular worked on the implications of an expanding universe, pioneering the attempt to unify quantum theory and relativity. And it was either they or Dmitry Blokhintsev and Fyodor Galperin who proposed the particle that would be the quantum of relativistic gravity, the graviton. Also part of this effort was a young George Gamow — but more on him later.

Later on, Soviet physicists did object to the Big Bang theory, for both good reasons and not-so-good ones. First, it was the not-so-good ones. Just as the Western scientific community was starting to come around to Lemaître, his ideas fell out of fashion in the USSR. As you seem to have already considered, and as u/restricteddata discusses in this thread about the theory's reception in Maoist China, they were seen as being too Abrahamic, and it seemed to contradict Stalin's understanding of dialectical materialism.

In 1938, even, Bronshtein's continuing defense of Friedmann and Lemaître, among other things like his work on quantum relativity, led to his execution. So you could say it was a result of different reactions between the philosophical realm and the world of physics and astronomy, and you wouldn't be wrong, but I don't really think that categorizing the theory's opponents as pure philosophers and its proponents as pure physicists is very helpful either. Science is always philosophical, and philosophy is always scientific, and you can't always separate them. (Like the continuum of literature to art to architecture to urban planning to politics on the Moscow Metro.)

Now, the development of cosmological theory continued in the West, but even there it wasn't entirely settled in favor of a Big Bang universe until the 1960s — Fred Hoyle was holding onto the steady-state theory until at least 1950, as u/restricteddata discusses in this other answer about the name of the theory. However, the Big Bang theory eventually won out, and that George Gamow I mentioned earlier, having emigrated to the US in 1933, made a key contribution. In the 1940s, he showed how the Big Bang must have been "hot" — that is, the extremely early universe must have been mostly radiation rather than matter. Then, in the 1960s, Arno Penzias and Robert Wilson discovered cosmic background radiation, which both suggested that a Big Bang had to have happened and that Gamow was right about how.

Now, though, Soviet physicists began to object again, and this time for some pretty good reasons. In the 1970s, Soviet physicists Andrei Linde and Alexei Starobinsky began to argue that the "hot" Big Bang model was insufficient and needed a very specific set of initial conditions to work out the way it did. Working off of their ideas, the American physicist Alan Guth proposed in 1979 that the universe must have experienced a period of extremely rapid inflation just after the moment of the Big Bang, and that this way, the very specific conditions were no longer necessary.

I really don't understand the argument much better than that, but the point is just that it's not as simple as the Americans getting it right and the Soviets getting it wrong — there's been a process of development that slowly got us the Inflationary model we have today, and at different times, different people on different sides of the Soviet border have had greater and lesser hands in developing and promulgating it.

(Also, Inflation theory has also come under some criticism, but only after the fall of the USSR, and for very technical reasons that I don't understand at all, so I'm not going to discuss it. I just say that so that you don't get the impression that it's the be-all-end-all of cosmology, which is still a developing field.)

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Sources:

Graham, Loren R. Science in Russia and The Soviet Union: A Short History. Cambridge: Cambridge University Press, 1993.

Guth, Alan. The Inflationary Universe: The Quest for a New Theory of Cosmic Origins. London: Jonathan Cape, 1997.

Kojevnikov, Alexei B. Stalin's Great Science: The Times and Adventures of Soviet Physicists. London: Imperial College Press, 2004.

Tropp, E. A., V. Ya. Frenkel, and A. D. Chernin. Alexander A. Friedmann: The Man Who Made the Universe Expand. Cambridge: Cambridge University Press, 1993. (It's kind of funny, I think they're trying to emulate Friedmann. As they say, "[his] book was not that of a popularizer" — and nor is theirs. Sheesh. It's half-incomprehensible. Equations up the wazoo. I didn't sign up for this.)