Process of elimination starts by family, background, then you get to class. Just skip to the end for the tl;dr answers.
Cadash aren't really a good fit story wise. Sure, lyrium smuggling sounds like a good connection, but in reality, the Carta don't play a major factor in the story. Therefore this eliminates the Dwarves. If you do like to play this race, the class that makes the most sense is Dual Wield followed by the specialization Assassination, because they wouldn't want to be well known before becoming Inquisitor, but to those that do, be a being that commands fear. Being trained as a rogue would give them the necessary background to be able to command the Inquisition and make those hard choices, as well as be a prime negotiator.
The Trevelyans are a good candidate. While the mage class is intriguing, Trevelyan mage wouldn't be trained in the ways of being a noble. As the mage class is taken from a young age, they would have no training necessary to step into the role of Inquisitor. Therefore, following the ways of the family would make the most sense. They would be pretty strong guided by the Maker (not necessarily the Chantry) and would be well trained in the ways of the Templar (also siding with them). While most templars are sword and shield, we need our hero to stand out. Two Hand would be the way to do that.
Lavellan is a strong candidate. Through they are sent to spy like Cadash, they likely would be sent as a diplomatic observer. Many diplomats also are spies and therefore, there would be nothing wrong with Lavellan being there. While a diplomat is usually reserved for a very well contributing member of society, a spying diplomat can also be a great way to gauge the worth of the next leader of that society. Therefore, Lavellan is likely the next in line to be the Keeper. I haven't seen a Keeper yet not be a mage, Lavellan would likely be a Rift Mage due to their ability to command of the Fade. Especially since I believe Solas says Lavellan is gifted at it.
Now you might've realized I've left Adaar for last. This is because I believe Adaar to be the true Inquisitor of Dragon Age: Inquisition. Adaar has been brought to the Conclave as a neutral party between the Mages and Templars. This would mean you are hired by the third party that is brokering the peace talks themselves: The Divine. This would give Adaar a perfect reason to be entering what likely was The Divine's chambers, as they would be hired by her and likely take commands only from her and her retainers.
Being a Vashoth also explains why the Inquisitor needs to acquire certain perks that for other races, they would already have. Trevelyan would have an understanding of military tactics and how to act as a noble. Cadash would have an understanding of money and the seedy side of the world. Lavellan would have an understanding of magic and medicines/herbs. Though the Vashoth do not believe in the Qun, this does not mean the Vashoth would forsake the efficiency of what to do. If Adaar's job is a mercenary, he would learn only things relative to being a mercenary. This means he has knowledge of tactics, money, delicacy, magic, medicines, etc, but not an understanding of them.
So what class makes the most sense, theme wise? Warrior. It is stated in the warrior or rogue class that Adaar, Valo-kas mercenary group as a soldier who has made a name within the company over the years. This would likely put Adaar in line with being next in command for the mercenary group and this being a proving ground. Though Addar would not be as well trained, because Addar would need to prove his/her worth. The warrior class is favored over the rogue class, because to be in line for a command position, they would need to be a front type of person. At least, so it would seem for the Qunari and Vashoth. Being a mage does not grant any privilege of such, as the mage class is ostracized, therefore Adaar would not be a mage.
The stock armor the Inquisitor wears upon creation, Mercenary Coat, also helps this narrative. The only mercenaries that would be at the Conclave would be under The Divine's command. Cadash would be disguised as a servant, Trevelyan would be wearing noble's clothing or templar armor, and Lavellan would be dressed akin to a keeper or diplomat. The only one wearing anything like a mercenary would be Adaar who is a mercenary. Yes, the Mercenary Coat matches the same armor that many humans wear, but there's an easy explanation. Remember when Iron Bull has the Inquisitor met those two members of the army? To fit in, they wore similar armor. This would be the same thing as that.
With that out of the way, we can narrow down the class of warrior for Adaar. It would make a lot of sense to assume that Adaar is similar to Iron Bull and thus a two hand warrior. However, the mercenary group known as the Valo-kas would need to have a been a highly well known and large group, as they would need to be in order for The Divine to hire them to provide protection at the Conclave. If you are hiring someone to do protective work, you'd need someone skilled at protecting others as well as themselves. The best class of warrior to provide protection is the sword and shield class. The best specialization of warrior to provide protection is the champion. Therefore Adaar would be a Sword and Shield Champion.
Though I wasn't meaning this to be a "here's who the Inquisitor is" type of post. It is going to come across that way and I apologize. This is how I came to the decisions I did for each and why I believe the Inquisitor to be the class I claim.
For those who jumped down to the TL;DR portion.
Cadash: Dual Wield Assassin
Trevelyan: Two Hand Templar
Levallan: Rift Mage
Adaar: Sword and Shield Champion
My Choice: The Sword and Shield Champion, Adaar