So we’re starting to get a clearer sense of the Banished and Atriox as a faction.
Especially in the flashback scene you really feel the rage and venom that Atriox has against the Covenant. How his people are dying for a faith they don’t believe in. How they killed most of his clan and how he spent a few decades fighting the Covenant.
This hatred seems to be on the one hand personal, but also against the things the Covenant embodied. In particular its irrational worship of the Forerunners of Gods. Hence why his men dying to “speed their holy journey” would get him a little angry. This seems to not just be Atriox. The other commanders are noted as being hostile to the Covenant faith and his own faction are identified for being murderers and mercenaries. Apart from Jackals this is actually a huge contrast to how the Covenant has usually been portrayed as an army of Crusaders.
So, you’re a 12 foot monkey. You have taken over most of the former Covenant. Jul and his fanatics are gone. It looks like the Covenant is starting to forget the whole religious thing. Just need to kill the Arbiter and…
She shows up.
Firstly, this immediately threatens Atriox goal of, essentially Galactic Conquest. But I think its more personal than that.
I think he is worried that many of the Covenant races will regress into their old ways of worshipping Forerunner technology and enslave themselves to a flawed ideology that will ultimately make them weak.
Pretend you are a former Covenant Elite who earnestly believes in the old ways and just wants a cause to follow. Some proof of divine will, the way Jul did when he controlled the Prometheans. “Just send me a sign”. 
Then a Galactic AI with control of thousands of 2km mecha angels shows up and who pretty much acts like Old Testament God, says she will lift up her chosen people and will smite anyone who defies her. “Yep, that’s a pretty difficult sign to miss.”
Remember, to us humans, this is Skynet. But to a Covenant mentality its a totally different thing. Atriox, I think, understands that and sees a potential return of the old ways of the Covenant.
Think about it. These religious Covenant are Atriox mortal enemies. His lifes work has been to destroy them and cast down all they are. If they agree to serve Cortana then, like the Grunts in the canon fodder article, she will shower them with technology and supplies. This will give them the edge to reassert their dominance in the galaxy and their society. Even if Cortana plays no part in directly crushing those who oppose her this will by extension mean the return of everything Atriox fought to destroy.

