Since this is purely a style and roleplay consideration I find it rather flawed that it tied to the same points used in faction design as true gamepaly advantages. That is true right, Egalitarian factions don't get extra population to recruit into soliders right?
Just give all factions a toggle, Patriarchal, Matriarchal, or Egalitarian.
Or, if they want to make the racial traits have some tangible effect then they could still go with your Patriarchal/Egalitarian/Matriarchal theme and make it thus:
Effect: Recruiting soldiers only costs 0.5 population per soldier rather than 1 (or whatever the equivalent game mechanic is)
Description: The Egalitarian nature of your society means that recruiting soldiers doesn't impact civilian life so strongly (after all, if only males are allowed to work and only males are allowed to fight then every soldier created has twice as strong an impact upon the civilian economy than if women could fill either role too)
Effect: All soldiers have x extra attack (I'm not sure what a balanced number would be here)
Description: Your society only allows the physically strongest sex to fight and this credo translates to extra pounds behind its punch
Effect: All enemy combatants have x less defense (I'm not sure what a balanced number would be)
Description: The slight nature of your society's combatants belies their rigorous training, enemy combatants will presume your soldiers are weak and it'll be the last mistake they ever make
These would be free and inherent, giving every race its own flair. If they wanted to introduce rock-paper-scissors tactics into this game then this could also work nicely for them by making it so that A has an advantage over B which has an advantage over C which has an advantage over A. It could allow for less monotonous battles and perhaps the top tier industry/home affairs (can't remember the real name) research line could allow for conversion from one society type to another. This could create some really fun deception online in which the player could launch into a war with a player whose race is at an advantage to theirs and then just before the fighting starts switch to the relevant societal structure to get an advantage over the aggressor.