Countering Priest-heavy civs without Chariots

I was playing an 8-player FFA RM last night as a random Civ and got Persians. Among my opponents were 2 Phoenicians and 1 Egypt, and unfortunately they were far enough away that I couldn't Cav rush them right off the bat (other AIs in between). As the game went on, they all started going pretty heavy on Priests, and was losing units left and right since Persians can't build Chariots. When fighting a Priest-favoring civ in the late game, is there an optimal way to deal with them? My main plan was to try and use a single Catapult to snipe them (still outranged though since I had 12+2 and Egypt gets 10+5) and instantly convert it back if the Catapult was lost; or to send a few Scouts or Villagers in to soak up the Priests and use Cataphracts/HA to take down the Priests as they regenerate. Is there a better way to deal with Priest-based civs in the late game (besides the obvious "don't let them get Priests")?