The usual Catholic objection to “A Mighty Fortress Is Our God” was not so much that Luther wrote it, as that it was used as a battle and slaughter song by the Peasant Revolt that Luther instigated (albeit not on purpose), and by several of the Protestant warring factions in the brutal wars afterward.
So baaaasically, we’re talking about singing Nazi songs in a synagogue, or at least, singing “Marching to Georgia” as a US patriotic song… in Georgia. (Except that Sherman’s troops didn’t do hardly anything objectionable, compared to the Peasant Revolt guys and the European wars doing horrible stuff to Catholics and other Protestants.)
Of course, the traditional objection by Catholic church musicians is that one of the verses actually ends by saying that the devil has no equal on Earth, making the song a hymn to Satan unless you go on and sing the next verse after it. (And yes, the next verse is about Satan’s butt being kicked; but the Satanist version happens fairly often in churches where the music director isn’t observant enough.)
Some people have associated the song’s praise of Satan with the historical bad stuff done by its historical singers, which is at least worth considering. (And with all the good stuff that gets cut out of modern Catholic hymnals, I don’t see why they didn’t cut the controversial Satan verse too!)
And no, they don’t ever teach us this kind of useful stuff in history classes… which is why our contemporary Catholic generation doesn’t realize there’s anything objectionable about the song except Luther’s name on the lyrics.