While I know the power of Banks, believe me I work at one.
I am actually reminded of a Scene from The Dark Knight Rises
https://www.youtube.com/watch?v=rY7stDTWjRI I Paid You are Small Fortune, And this Gives you Power over me?
Money, Banks and Authorities doesn't count for much in the face of True Belief.
The Tyrells/Iron Bank is meaningless to a Believer like the High Sparrow.
Very true, but lets not forget about the Faceless Men. Who do they work for? Would the Iron Bank hire them to assassinate people who get in the way of getting their money back? Probably wouldn’t influence the High Sparrow, but it could possibly be a way to get rid of him if he proves to be too bothersome.