The Ben-Hassrath are not job checkers. They evaluate if someone is fulfilling their assigned role in the Qun. If the person is not, that person is reeducated to continue fulfilling that role, or be eliminated. Iron Bull, for instance, isn't given a new career choice. He starts off an agent for the Ben-Hassrath, is reeducated when he gets burned out, then continues his job in the Ben-Hassrath.
And again there's no checks and balances independent of the power structure. Nobody is available to make sure the Ben-Hassrath are doing a good job or the Tamassran are doing a good job - or that anyone fulfilling their roles in those structures are happy with those roles.
The Arishok lays it out plain in Dragon Age 2. You are free to make a choice in your role. You either accept it, or you reject that role, then are killed.
A faction of the Ben-Hassrath are re-educators, a faction of the Ben-Hassrath are on the lookout for those Qunari who are failing in their role in the Qun, and bring them in to be reducated. By those role descriptions, they are job checkers.
While you are correct that Iron Bull isn't given a choice for a new career, he is given a new career. Iron Bull was a Ben-Hassrath who functioned as a police officer. Got burnt out, and was given the new function of spy. It's a new career, and presumably, a new job title(because it seems a bit weird to me to have what is essentially a police officer being given the job title of "liar"), but his role, as a member of the Ben-Hassrath, of the priesthood, was still the same.
And I'm not entirely certain what you mean by "no checks and balances independent of the power structure." Just because you aren't told of their existence doesn't mean there are none... As I understand it, "a good job" to the Qunari means "succeeding in fulfilling the role within the Qun". And that would mean the Arishok, Arigena, and Ariqun ensure those within their respective branches are doing "a good job." If a Tamassran continually assigned "substandard" Qunari it's unlikely it would go unnoticed by the Ariqun(who is in charge of both the Tamassrans and Ben-Hassrath)...
To follow up on my earlier point about the permanent assignment of roles by the Tamassran...
In Dragon Age 2 the Arishok says:
"Karasten are soldiers. The Qun made it so. They can never vary from that assigned path, never be other than they are meant to be. But, they are free to choose within that role. To accept and succeed, or deny and die. Glory is clear and defined. It is an undeniable certainty. What "full advantage" can you take without that authority?"
Hawke replies: "He's free to choose to obey? That isn't contradictory to you?"
The Arishok replies: "He chooses to be. As do we all, long before any of your meaningless freedoms are presented."
This is a good clip, because I believe that it shows the fundamental idea that you aren't grasping: The "assigned path", the role, that the Arishok is discussing is that of a Soldier, not of a Karasten... A Tamassaran doesn't assign a Qunari to the role of Karasten. The Tamassaran assigns a Qunari to the role of soldier. If a Qunari is assigned to be a solider, and they can choose within the role, accept and succeed, be it succeed as a Karasten, an Ashaad, an Arvaraad, a Sten, or even an Arishok. But if a Qunari finds themselves burnt out as an Arvaraad, that doesn't mean they cannot become a Karasten(provided they have the skill-set)... But a soldier cannot become a crafstman or a priest, because that is not their role.