If you want to preserve monarchies and nobility, you can include bloodline entailed magic (ala sorcerers or some forms of warlocks) or you can simply make a convincing case that the Sword Bros can handle themselves. Your real issue is that in having a god of bureaucracy means that the government will always be much more similar to Confucian states, which have centralized governments vaguely like our modern ones, than to Western Feudalism, which is basically confederal warlordism. But often fantasy settings more resemble this anyway, since the writers are moderns native to the modern city state!

I think also you are underestimating the chaotic gods. The whole history of civilization until very recently has been the slow extinguishing of human's natural way of life and its replacement with state-forms born of sedentary agriculture. It was really not that long ago, even in incredibly densely packed Europe, when one could "head for the hills" and join a society that lived in tribal anarchies as humans have for ten thousand of years. Even into the eighteen hundreds, people who were kidnapped into Native American tribes often "went native" whereas the reverse was almost never the case (and the kidnappings did occur in reverse). Such societies would clamor for song, rain, and freedom. (Probably not animal husbandry though) By contrast, a world that's mostly truly conquered territory, that's bent either to agricultural or industrial purposes through human(oid) engineering is also one that's probably on the verge of mass democracy (there's no vent for the pressure states impose and the crowding makes states more dependent on cooperation). And in a mass democracy (or even a state negotiating avoiding that, as pre-19th century Britain or Bismarck's Germany), gods that center freedom and independence (like Zarthus) will be very popular. So it's only in relatively narrow circumstances (early age of exploration, games modeled on the American expansion into the West) that I think the chaotic gods would be genuinely on the outs with humanity.