I think we’ve got something here. A first attempt by Jules, drawn from the comments:

Patriarchy was in trouble, what a sad sad story
Needed a strong leader to ensure its future glory
Where oh where was he?
Where could that man be?
We looked around and then we found
A prop for the papacy–
And now it’s…

Springtime for Constantine, and patriarchy!
Did you think he wasn’t involved?
Dictating things catechetical
Look out, now you’re heretical!

Springtime for Constantine and patriarchy!
Winter for women and gays!
Springtime for Constantine and patriarchy!
Come on common folk, go into a trance!

I closed my eyes to Jesus’ fam
That is why they call me Sam

Don’t be stupid, be a smarty
Come and join the Vatican Party

Springtime for Constantine and patriarchy!
Conspiracy as big as your head
Call out our elite ninja monks
Make them albino for good luck!

Springtime for Constantine and patriarchy!
Men are in charge once more!
Springtime for Constantine and patriarchy!
You won’t expect us at your door!

Go for it, folks!