Hold on, if it works for the circle FTO it also does something with the normal FTO:
The point is the same as with a normal 5x5x5 solve. You do a slice move, than you do 4 face moves (X Y X' Y') and than you only undo the slice move. Since the pyraminx party can stay scrambled the whole time and is just solved once in the end (just like the 3x3x3 part on a 5x5x5 cube solve), you actually should be ably to spare all those moves that restore the pyraminx. In other words some face moves.
I am not sure about the pairing up of three edges with three triangles though. You need four groups of those on non adjacent faces. Try it and tell me more about it.