One issue of super national organizations is "Quis custodiet ipsos custodes?" (Who guards the guardians?). The problem in the setup is this is not just a super national organization like OCHA, but one with massive material and martial resources available.
Over the years, this will breed two responses. Various nations and groups will be appealing to the organization for assistance against whoever they perceive to be their enemies, and of course they will be using everything from persuasion, bribes and even threats of force to attempt to enlist the order to bring their resources to the issue.
The other response will be jealousy and resentment that the Order are not bound by the same rules as everyone else, and that they have real or imagined access to powers and resources beyond that available for ordinary mortals and kings.
In the real world, this might be close to the situation that developed as the Poor Fellow-Soldiers of Christ and of the Temple of Solomon (Knights Templar) gained military and financial power over the course of two centuries. Eventually, they fell under the jealous gaze of King Philip the Fair, who managed to seize the members of the order and attempted to seize their wealth as well. (How the money escaped is not clear today, although one theory is the Knights operated a banking system similar in conception to today's and the "wealth" was in the form of accounts receivable and loans rather than the piles of gold coins King Phillip seems to have imagined).
Given the swirling currents that will surround the Order in your setup, I suggest that the most likely outcome after centuries of receiving requests, bribes and fighting attempts to seize their wealth and power, the Order will have become somewhat inward looking and to all intents and purposes have been corrupted. The Grand Master will certainly intervene on your side of the conflict, if you can come up with the proper price. What the proper price for an organization like that is up to you, of course.
The other outcome is their growing power and hubris will eventually call down the wrath of the gods, much like the Valar caused the downfall of Númenor (recounted in the Silmarillion ).