No modding here, just a claimant war followed by a catholic invasion.
Long story short: the byzzie under a crappy basileus got stomped by Muslims, as usual. So I invited the brother of the basileus, demanded him to covert to Catholicism, and push his strong claim on byzzie. Success! And the new basileus was as crappy as his deposed brother. In Rome, the pope was abhorred by the news that a hedonist, lustful, gluttonous cynic ascended the throne of a "major catholic power". When I asked a permission to invade, the Holy Father happily granted his blessing after just 3 days of consideration. So it began.
After the dirty works, the Ecumenical Patriarch and the varangian guards became my new vassal. I tried to convert the Patriarch to Catholicism for giggles, but all attempts failed:
Finally, I gave up. It seems the religion of the heads of religion is indeed hardcoded. You cannot use court chaplains to convert them, demand their conversion, or use console to change religion when playing as one of them. Then the old Patriarch died of natural causes and succeed by a nobody. I paid no attention to the Ecumenical Patriarch again until one day...
Well, that's funny.
To the OP:
Since Iconoclast has no head of religion, you can't get the necessary invasion cb. If you want to use the method described above, you have to mod the game as Velorian suggested.
Edit: Beyond my wildest expectation, the new Ecumenical Patriarch was a super badass, he captured booth Seljuk sultan and the Sunni caliph during the 2nd crusade for Jerusalem.