Integrity literally means to hold together. As a virtue, it suggests that your beliefs and your behaviors remain the same over time - that you do not vacillate or try to put on a different face in different circumstances. What does that have to do with Koshiki no Kata?
Koshiki no Kata is intended to be performed with solemnity and dignity, but how to be solemn and dignified is pretty much left up to the practitioners because it can't be taught or affected. Do you pretend to be a Samurai and feign disdain for your uke who is obviously beneath you? Do you move extra-slowly, refusing to break your upright posture? What do you do to make your kata look dignified?
Well, if you have integrity, then you do absolutely nothing special. If you already have a deep personal dignity then all you have to do is let it shine through the kata. You don't have to put on a new face or affect some different behavior to pretend to be dignified because you already are. Perhaps that is part of why this is considered such an advanced kata - it just takes a goodly part of a lifetime to develop that sort of stable, mature self-image.
If you have to try to act like some old, dead, Japanese murderer in order to feign dignity then you lack integrity.
Watch the Yamashita demonstration of Koshiki no Kata again and notice how naturally they walk and stand and move. Nothing is affected in their behavior. They are just being Koshiki. That is both dignity and integrity.