If this is state employment, which is what it strongly sounds like, you are fighting a terrifically impossible battle to get anything much changed. The departmental HR is in complete control of the top tier execs, and they have some leeway to create and change positions, but you know what? Unless you are terrifically politically connected at the state level, they won't. Your department will have an EEOC person, and there is always the overall state EEOC office, but you're pitching the job as a "different compensation for gender related reasons" and that dog won't hunt. This is just a trap that you've fallen into with this particular position, in this particular department. You're doing a lot more than the traditional assistant, you know it and they know it, and the only way to get a raise in salary is to change jobs, change titles, ergo change pay grades. They can overcome handily by showing that yes, sure, they'd hire a qualified person of the opposite sex for a similar position anytime.
A large organization of any type is likely to resist change and the folks in HR are likely wise to duck opening Pandora's Box as to if some job classifications are problematic along any sex and pay lines . OP just because you are doing a lot more than job description requires does not make it a gender discrimination issue..
As an example my daughter does far more than her job description addresses in a very very large organization , the grade of the position is not going to move in that big institution , nobody is going to tamper with it. However her boss sure found a way to get her paid more , comfortably above the supposed cap on that grade.
IN some other large bureaucratic organizations I am well aware that's it's easier to create a new position and get it graded higher than it is to change an existing position .
One not so young female acquaintance who felt "trapped " in lower paying "admin " role played her seniority status to bid into a very different typically male role at a much better pay rate ...in an organization where seniority counted a lot.
This is a whole lot about playing the system where you are . ...and making your own breaks .