I might be going nuts and I'm hoping someone can confirm either way but do you get higher FATE rewards the higher rank you are in your Grand Company? It seems that way to me.
Honestly not sure if GC plays any role in it, but it certainly could. However, you certainly get more rewards the closer you get to the actual level of the FATE which is more likely what's happening.
Level 30 doing a 35 FATE may see less rewards than 34 doing the same FATE.