I'm not sure of the exact source, but you could narrow your search by looking at other Rai documentaries on the war such as this one. They're in Italian but from what I gathered most of them are free once you make an account.
I just read the rest of the tweet thread, and it appears to be from this soviet documentary from 1939. Kinda impressive how the most accessible archive is from an Italian sub.
Thank you. In this video appear this song and others of the Spanish Civil War. I appreciate you found it because appearse on the video the original version of the song, no other versions that were made after the conclusion of the war.