
Mysterious Stranger |

You could probably make it work with a few other classes, but inquisitor is an incredibly good class for this concept. When a player mentions they want to play a <insert role> of a specific deity a divine spell caster is usually going to be the most common suggestion. An insurance adjuster is probably going to need a lot of skills and most divine casters usually don’t get a lot of skills. Other than inquisitor the only divine spell casters that get decent number of skills are usually nature based, which does not really fit either the role or deity you want.
If you want something less obvious it is not going to be a divine caster. A bard could work with the right archetype. Either an archaeologist bard or a detective bard could do well as an insurance adjuster. If you want to play a non-caster an unchained rogue could work. The phantom thief archetype gives up sneak attack, but gets accelerated skill unlocks. You can get the 5-rank skill unlock at 4th level, the 10-rank skill unlock at 7th level, the 15-rank skill unlock at 10th level, and the 20-rank skill unlock at 14th level. Being able to use sense motive to read minds would be very useful for an investigation.

Melkiador |

I guess this leads to some other questions. How religious should this character be? Sure he works for Abadar, probably even respects Abadar, but does he worship Abadar? Or does he get his class abilities form Abadar?
How good is this character supposed to be at this job? Just because he's an insurance adjuster doesn't mean that this is his main focus in life. Maybe that's just how he pays the bills while trying to pursue his real dreams.

Kasoh |
Investigator, Inquisitor, Rogue, Bard, or Diviner Wizard. Being an insurance adjuster means that you have to be able to discern the truth of a matter and have a sense of fair value of what has been lost. You need skill points or magic to get that done. If they're an 'expert' or used for 'special cases' maybe they need a sturdy combat chassis as well in which case a slayer could probably do well.

Wonderstell |

Knight of Coins Half-Orc with the Overlooked Mastermind and Skilled ARTs.
Put your FCB in skill ranks and you'll have 6+Int ranks per level as a Paladin. Your Sense Motive is buffed by the Overlooked Mastermind ART and the relevant Blessing of Prosperity.
At lv 5, dip a single level into Bard for the Pageant of the Peacock bardic masterpiece and you're an extremely efficient skillmonkey paladin.

Mark Hoover 330 |
Wow. So much exploded in my brain with the OP's question. An Abadaran Insurance Adjuster...
Well, looking up an article on what RL insurance adjusters do, along with the interview/assessment portion of the job they need to be able to settle claims quickly and amicably; this suggests ranks in Diplomacy and maybe a Profession skill as well. They work closely with lawyers as well as law enforcement for info gathering and document preparation. Many also conduct their investigations in person, in the field, suggesting the agent needs to be physically capable/endurant of travel.
Putting that into the mechanics of the game, you'd want to pick PC's with class skills and abilities or magic around these areas. Adding Abadar worship into the mix is actually trivial unless the OP specifically wants that to be a feature of the PC's abilities as well. Just looking at the job description though, you either use spells or:
Diplomacy, Sense Motive, Perception and Profession as Class skills
Decent Wisdom and Charisma scores for the skills
Some investment in Con for travel endurance requirement
Linguistics or a high enough Int score to speak multiple languages
As for what class this all fits into? Now that REALLY depends more on how you want your PC to resolve combat than their chosen profession. I'm gonna go alphabetically here:
Core classes
Barbarian: the Urban Barbarian gains Diplomacy, Knowledge (Local), Linguistics and Profession as Class skills while retaining Perception as well. If you pick up Sense Motive through a Trait you've got the skill components down. Obviously you'll have the endurance for travel and Controlled Rage allows the PC to still use Cha based skills while raging.
Bard: SO many options here I won't go through them all. This class would be an obvious choice
Cleric: if you're going to focus on Abadar worship as part of your PC's identity, a Wis and Cha based cleric is a decent choice. Outside the vanilla class there's the Cloistered Cleric to represent a more bookish, office-bound claims adjuster, the Exalted or Evangelist boons for a Divine Paragon would help so long as the PC was in a city of 5000 or more, or obviously choosing Inquisitions vs Domains could work
Druid: one of Abadar's associated archetypes is the Urban Druid which could easily support the concept of a claims adjuster in the deity's service
Fighter: While it might feel counterintuitive to play a fighter for this PC, you could actually pull it off by putting a secondary emphasis on Int and being Human for extra skill ranks or Half-Elf for a free skill focus. Beyond that, the archetypes are more about how you want to deal with combat so...
Monk: just like the fighter, but easier. The vanilla, Unchained class gets Perception, Profession and Sense Motive as Class skills as well as 4+ Int mod ranks/level. The concept of the class, that they learned fighting techniques and philosophies can tie them into Abadar on a monastic level, which also ties back to the Lawful nature of both the deity and the class. After that, the class ability choices and archetypes are more about how you want to fight
Paladin: like, nuff said. Abadar's paladin is a Knight of Coins; they add Perception and Knowledge (Local) to their Profession, Diplomacy and Sense Motive skills, Eye for Forgeries lets the paladin suss out insurance scams perpetrated by magic, and imagine how well loved a Knight of Coins would be if, as part of settling claims, they were able to Lay on Hands and deliver a boon like letting the claimant get an extra 10% on items they sell back for scrap or whatever?
Ranger: the Urban Ranger is another Abadaran archetype. They've got Knowledge (Local), Perception and Profession as Class skills, so you might need to supplement with Traits or magic. In a favored settlement your Knowledge (Local) improves; this type of PC is more like the kind of claims adjuster well known to work with local law enforcement, lawyers and such to resolve claims
Rogue: like, are you kidding me? Like the bard, this is an easy fit
Sorcerer: focus on magic to settle claims
Wizard: focus on magic to settle claims
That's just the Core classes. I don't have time to do all the other classes and likely no one's reading this far anyway. Still, my point is only that there are MANY ways to skin this particular catoblepas. Happy adjusting!

Mysterious Stranger |

The skills that an insurance adjuster is going to need are Diplomacy (negotiating), Perception (observation), Sense Motive (detecting lies), Knowledge Local (law), Appraise (determining value), Linguistics (contracts and forgeries), and Professional skill Barrister (lawyer). Using traits can turn some of them to class skills. In addition to this the following skills will be highly useful Bluff, Intimidate, Spellcraft, and other knowledge skills.
For stats they will probably need a good WIS. Some things are not covered by skill and the GM may call for straight stat rolls. Having a low WIS will mean you fail those. A good INT is also going to be needed because it gives more skills, and a lot of the needed skills are INT based. A good CHA will also be useful, but if the character is an inquisition, they could take an inquisition to use WIS instead of CHA for some skills.
In a world where magic is real having access to magic is going to be important. The character might be able to get by using items, but that is going to be expensive. Divination magic is going to be the most useful. Combat spells are not going to be as important to the role but may be needed to round out the character.
The character should also have a good will save to be able to avoid being influenced or read. A lot of divination skills like detect thoughts grant a save. If your opponent is able to learn what you know they have a huge advantage.
Also look for class abilities that are helpful. The inquisitor’s stern gaze gives a bonus to intimidate and sense motive. It is probably the single most useful class ability outside of a high-level divination spell. Intimidate can be used to force a person to tell you what you need to know. Sense motive can determine if they are lying to you. The combination is hard to beat for an investigation-based character.

Mark Hoover 330 |
So, has anyone thought about Psychic as another "ideal" class? They're only 1/2 BAB but 9th level psychic spellcasters. While they only get 2+ Int mod skill ranks/level they have access to Diplomacy, Knowledge (Local), Linguistics, Perception, Profession and Sense Motive as Class skills. Enlightenment and Rapport leap to mind as useful Disciplines, but maybe there's others that'd work. Just figured I'd throw it out there as well...