I thought of something, guys...
What if the Nimble Fabric isn't from Nitori, but from Miko?
Miko can teleport with her cape, so it could be that Seija just stole that.
Of course, that would mean that Miko may end up appearing in the game herself, and with her, some of her allies too, with the intent of "punishing the filthy youkai who would dare steal from the prince"...
Also, on a separate note, I believe that the items shown are NOT the only items that Seija might use in the game. For example, I recall that Sakuya in the fighting games could throw status fields that stops whatever touches them. What if Seija had stolen the Luna Dial, and using it allows her to fire off a status field in front of her, freezing the bullets that it touches while it persists and thus creates an opening in the impossible danmaku. Interesting idea, yes?