I don't think the OP is trollish. Those are legitimate beliefs in the OP, not exaggerations. There were similar arguments for and against same-sex relationships. The outcome of that was Toady implementing the bare minimum of new tags in the raws such that the same-sex behavior is possible. That's the most likely outcome here, too.
It looks like the Navajo and Hindu examples are actually every bit binary on the gender levels. Just the Navajo decoupled gender from genitals, but they drew clear boundaries between allowable masculine and feminine roles, including marriage. Modern Iran does something similar: they are strongly anti-gay, but pro sex change. As long as you have people of gender A marrying people of gender B, they don't care what your birth genitals were. If Navajo had had access to sex change surgery would they have done the same? Probably, yes. I don't see it as a clear sign the Navajo were extra-progressive, it's just a slightly different take on enforcing rigid binary gender roles. The people who say that Navajo had "4 genders" are basically making a mistake that contradicts their own general position: gender is social, sex is genitals. Having a word for a dude in a dress isn't proof that they have an "extra gender". We have words for that, too. If they are thereafter treated exactly the same socially as a woman, they have the woman gender.
As said above: dwarves really are mono-gender, with the only trappings being the bare minimum for mammalian procreation.
Modding in Navajo Genders in current system
Right now, any creature token that can be modded at the caste level could be altered on a per-gender basis. Then, you could make 2 male and 2 female castes and give each one opposite gender-related tags. Presto "masculine" female dwarves and "feminine" male dwarves. This wouldn't properly model the Navajo system though, because dwarf sexuality couldn't be constrained to only doing opposite genders. But SEX + ORIENTATION tags can actually restrict mate selection in a way similar to the Navajo! So the "fix" for Navajo modelling would be to use SEX tags as socially assigned gender instead, turn OFF same-sex relationships, and then apply the baby-making tags to whichever castes have the female "sex" (which can now be different to the SEX tags since we have stolen this to indicate gender). Chuck some description tags in there and presto, your own Navajo! The only weird behavior would be a pairing of masculine female/feminine female could still have children, which isn't too bad if you assume they had a donor.
Appropriating the SEX tag as gender instead has a lot of potential, since Putnam already showed it has nothing to do with whether and individual can bear children. Right now as far as I can see the only functional purpose of that tag is mate selection.
The only issue I have is with the word "procedural" in the OP. The OP has given some examples of some societies which either do or don't decouple the sex binary from the gender binary, but he/she hasn't given much if any examples of what sort of things this "procedural" system is going to model. If you give specific examples of how that could work as a system of variables then it might be more feasible. Just saying "procedural" doesn't do anything if you haven't shown a set of variables that could be adjusted in a sensible way.