The league was formed by a group of Paladins who have seen the influence and suffering that religion, nationalism and magic has bought the world and thus they disdain them all, especially magic.
The League believes that the only constant force that all respect (whether through greed, influence or economic sanctions), is money and trade. A Chellish Hellknight and a Paladin of Erastil both respect and are beholden to money.
By making Qadira the strongest faction, and amassing their influence and wealth, real global change can be effectuated. If managed correctly by "Good" people, such as those of the League who gain sufficient fame/rank/influence, it can bring a lasting and sustaining peace to the world.
To that end, Paladins of the League are always keeping in mind the "Greater Good" of the world rather than the individual deeds that may seem out of place to a LG Paladin. Quite often you will hear a Paladin of the League say, "No more band-aid solutions".
Therefore, these Paladins do not worship a "false god" or pay heed to organised religion. Instead they follow a philosophy of "Greater Good".
No "magic": Potions can be "alchemical", +1 items are "reinforced"
Silver adornments: Gold is the metal of greed and corruption.
All members may wear a silver signet ring (5gp) with the League logo emblazoned upon the face.
The League is technically open to all Qadiran PCs though Paladins and Cavaliers are most common.
Spell-casters are not welcome.
Those Paladins of sufficient level to gain access to abilities that replicate spells often refer to them as "acts of focus" and refrain from using those that have obvious outwardly-appearing effects. Most Paladin "spells" are actually quite subtle in their appearance.
eg. "Bless" is simply the "rightness" of the Paladin being focussed through their weapon - all good. "Create water" - not so good.
There are no "rules" changes in anything listed above. It is 100% flavour.
Golarion really doesn't do atheistic, maltheistic, or dystheistic paladins. They're drawing their power from a divine source
Quote the guide:
Characters who do not receive powers from a divine source may choose to be atheists or to have no deity at all
It sounds like you're trying to find a way to get paladins/clerics to be able to follow the Prophecies of Kalistrade (based in the country of Druma). While yes they're a religious organization, they don't give any divine powers (as far as I know at least).
Yes, definitely works fine.
"As their powers are influenced by their faith, all clerics must focus their worship upon a divine source. While the vast majority of clerics revere a specific deity, a small number dedicate themselves to a divine concept worthy of devotion—such as battle, death, justice, or knowledge—free of a deific abstraction."
Given your wording, I could see it maybe, yea.
"Knights, crusaders, and law-bringers, paladins seek not just to spread divine justice but to embody the teachings of the virtuous deities they serve."
Yea, I just don't see that working based on the class as-written.
I cannot find anywhere where is says that a Paladin MUST follow a god. I can find references that go against that though:
Well, reading a bit deeper into the Paladin class:
Divine Bond (Sp): Upon reaching 5th level, a paladin forms a divine bond with her god. This bond can take one of two forms. Once the form is chosen, it cannot be changed.
RAI and RAW that means a paladin has a (singular) god. Their code of conduct might be influenced by different elements, as you stated, but that has been part of the Paladin motif since...well...I think the 1970s?
So what you're saying is that a CLERIC doesn't have to follow a "god" but a Paladin MUST?
"While the vast majority of clerics revere a specific deity, a small number dedicate themselves to a divine concept worthy of devotion—such as battle, death, justice, or knowledge—free of a deific abstraction."
While the base Pathfinder RPG rules allow for Paladins and Clerics without Dieties, I believe that Pathfinder Society Organized play rules do not. As you are posting in the Pathfinder Society Organized Play section of the messageboards we are assuming that you are asking in reference to Pathfinder Society Organized Play and therefore those rules must also be applied.
Bloodlust, as Kyle said, don't get discouraged. I think the idea is cool (my main character is a Qadiran cleric of Torag) and I love the spin of using control of trade to effect change for the better in the world as a motivator for LG Qadirans.
Unfortunately, changing class mechanics without an allowed archetype just isn't going to work in the campaign.
An alternative to using Paladins, would be to have the group be Monks with a mandatory dip into Cleric, or some other multi-class variation combination to mimic it, OR even have it have been formed by FORMER paladins who have now turned their zeel towards this avenue, eschewing thier former dieties for the greater good they can perform this way.