Bull betrayed us because if you sacrified the Chargers (or never did the mission) he is loyal to the Qun. Without the Inquisitor convincing him that he'd become the mask and that in his heart he was more Iron Bull than Hissrad, he remained an indoctrinated soldier that was prepared to followed orders without question because the Qun demanded it.
As Weekes points out in the interview, we can make him choose the Qun, why'd you think he wouldn't obey it?
Even if he wasn't part of the Viddasala's operation, his loyalty to the Qun meant that he would accept the order of a superior officer, even if the operation itself was skirting the bounds of what they normally did. Is it so unlikely that the Qunari wouldn't be willing to break some of the rules to eliminate all magic in the south, even if it meant using means they normally considered anathema?
What the Qun says and how the Qun is applied, as well as how the should Qunari act and how the Qunari do act, often end up being rather separate entities. But like all zealots, they are prepared to rationalise, twist and reinterpret their dogma to give them permission to do a distasteful thing under the auspices of the greater good.
Just because the Qunari officially disavowed the entire thing, we have no reason to believe that it wasn't an officially sanctioned operation from the Triumvirate. And if not sanctioned, they knew but were prepared to look the other way and disavow all knowledge of the operation and players involved, as most nations tend to do when they conduct black ops.
After all, it would be an embarrassment for the Qunari to ever admit that the Inquisition caught them en flagrante delicito, so much so that they might as well had red paint on their hands, which were resting in a cookie jar, while they were sitting on the privvy.
While the Arishok probably wasn't acting within sanction during the Kirkwall incident, I doubt that the Triumvirate would have necessarily cared if he'd succeeded and the city had fallen under his control. But since the Arishok failed to take Kirkwall, like Viddasala with her plan to assassinate the Exalted Council, the Triumvirate are going to spin the heck out of the situation to make it seem like it was a rogue faction at work.
Varric tells us everything we need to know about Qunari spin with how they reacted to Kirkwall;
"I believe the phrase they used was, "Let us never speak of this again"."