While Twist of Fate does not change the type of challenge that was initiated, you are still being held responsible for POW claim in a challenge initiated against you. I read the Agenda as caring about the claim type in challenges in which you are the losing player.
"Initiated against you" tends to be read as "in which you are the defending player." Crown Regent doesn't get around Across the Summer Sea even though the redirected challenge does not technically "initiate" the challenge against the Targ player. If you take a challenge for someone you Support, thereby becoming the defending player, you can drop LW-Catelyn into the challenge as a defender, even though the challenge was not technically "initiated" against you.
So the whole "it wasn't a POW challenge that was initiated, even though we are resolving the claim of a POW challenge" doesn't seem to track. We'll see if Damon says anything different, but it seems to me that the effects of these two cards (Twist of Fate & Brotherhood without Banners) combine rather than the agenda trumping the plot.