I'm going to try for the last time.
The cover story is flawed and not really plausible, because it doesn't align with the actions of the Fly Squadron. Plain and simple.
AND
Since it's actually a fictional story, Touka could as well say that they are splinter/renegade group of Ashinto, trying to make name for themselves. The detail about wanting to stay in shadows was pretty much unnecessary and counterproductive, because it will raise questions.
And he is totally trying to deceive the princess... Why? Because there is no better lie than the one that people believe is the truth.
I'll try again because I'm extremely patient -.-'
I checked the Light Novel, the Manga cut out some pieces of the conversation (or it wasn't translated correctly I don't know).
1. One Ashint faction wanted to govern the world in light, the others wanted to govern it in shadows.
2. Belzegea explains that the "Light" Ashint didn't let them go, but he doesn't explain what happen to them.
3. Belzegea intended to be hired as mercenary in Cattlea's Army (i think it was in an earlier chapter in the manga, but this chapter didn't bring it back, like the Light Novel does) and act in the shadow. But since there already was a battle, they didn't have a choice but to reveal themselves.
And all of this doesn't matter because :
1. Touka uses this Ashint stuff as a cover to hide his true identity.
2. The entire world thinks Ashint killed the Black Dragon Squad with curses.
3. They witnessed an Ashint member kills a high rank demon with his curses.
4. No matter the backstory he creates, they know his curse is not fake.