I’m not so sure you could divorce Communism from its atheistic elements, and if you did you’d no longer have a meaningful philosophical system. It is based in principle on atheism, and so to remove that is to remove something very much core to it. It’s not so much that a “theistic Communism” would be in principle acceptable, it’s just that making Communism non-atheistic would void enough of Communism that there’d be nothing left.
Now, if you get down to brass tacks, there have been attempts to meld Marxism and Christianity in the form of the liberationist theologies of more recent times. The Church has never found these to be acceptable, for they take the already unacceptable Marxist ideals and couch them in the language of the Gospel (and vice versa, as they attempt to portray Jesus as some kind of revolutionary ideologue); this kind of thinking runs counter to the Church because it reduces salvation to deliverance from oppressive social structures and says that sin exists as a largely societal reality. This ignores the truth, that Jesus did not come for the sake of ideology, he came to save all of us from our sins; he did not come to liberate us from some worldly oppressor, but to liberate us from the slavery of sin and the devil.
-ACEGC