So the dictionary, Wikipedia, Simone de Beauvoir et al are all trumped by something you pulled out of your arse?

I’m going to split my response up into a couple of sections, and the longer sections will have a TLDR at the end because I realized that I wrote way too much in response to a few of your points.

Dictionary, Wikipedia

As mentioned earlier, Merriam-Webster defines “trans woman” as a woman; however, this doesn’t actually matter, as the dictionary is not something to be used prescriptively. Words are defined by the socio-cultural context by which they are used. Similarly, to say a woman is an “adult human female” because “the dictionary says so” is fallacious reasoning by the nature of how the usage of grammar in the prescriptive sense functions linguistically speaking.

Wikipedia works in an identical fashion as any dictionary. Wikipedia cannot be used prescriptively. Wikipedia also defines “trans woman” as a woman. This does not prove that trans women are indeed women, as that argument would be logically fallacious; likewise, pointing to Wikipedia which states that a woman is an “adult human female” as evidence that this is the case is also fallacious.

TLDR: Prescriptive use of a dictionary (including Wikipedia) is a logically fallacious line of reasoning, and therefore holds no value when discussing the descriptive use of words.

Simone de Beauvoir

As for Simone de Beauvoir, her perspective on gender identity and the acceptance of transgender individuals can only be concluded in terms of inductive reasoning. This reasoning can go both ways, but neither argument for or against her alleged support of transgender people can be concluded with absolutely certainty; unless, of course, the scientific community has created a way to bring back people from the dead, in which case we could ask Beauvoir directly. That being said, here is my inductive understanding of Beauvoir’s writings:

Beauvoir would have no issue with transgender people adopting the social roles of the gender they identify with, and, effectively, becoming said gender; this is seen through her claims that there is no biological basis to being one gender or another, her writings expressing how the freedom of natal girls/women is restricted by the social role of “woman”, which is imposed on them by society, and her expressive support for the idea that males and females are socialized differently in a gendered society.

Beauvoir would likely support trans women adopting the social role of “woman” (and therefore, becoming a woman) because (1) males are not socialized to regard womanhood in a positive light (and therefore, the desire trans women have to become women does not stem from societal demand), (2) she argues that gender is not confined to any kind of biological reality (and therefore, the desire trans women have to become women have nothing to do with their state of being male), and (3) if transcendence from gendered norms is the end goal that Beauvoir desired, she would have no issue with trans women becoming women (as the desire to do so stems from the unrestricted freedom that males have and are socialized to hold, based on the previous two points).

Beauvoir would also likely support trans men adopting the social role of “man” (and therefore, becoming a man) because (1) she regards femininity as a restrictive force leading natal girls/women to be unable to pursue personal freedom, (2) Beauvoir often regarded female masculinity as a more fulfilling means to achieving personal freedom compared to female femininity (as femininity is imposed upon females by gendered society), and (3) if transcendence from gendered norms is the end goal that Beauvoir desires, she would have no issue with trans men becoming men (as the desire to do so stems from the need to break from restrictive femininity and pursue masculinity, which is the more fulfilling role in this gendered society).

TLDR: It can be inductively reasoned that Beauvoir would support the validity of transgender women and transgender men because of her belief that there is no biological basis to being one gender or another, her argument that males and females are socialized differently by a gendered society, and her writings which argue that that socio-cultural demand for females to conform to femininity leads them to be incapable of pursuing personal freedoms and fulfilling life.

Definition of “Gender”

The definition of gender as being a socio-cultural norms (gender expression) and social/gender identity is not something that was created out of thin air. It is a phenomenon that regards the socio-cultural and historical conceptual understanding of gender as the means that defines it. It is descriptive of what gender is, rather than prescriptive.

As for evidence that gender is a combination of gender expression and gender identity, we can look at the mass social movements that are happening right now. In the US, we have Black Lives Matter protests advocating for police/prison reform, legal protections for women, and social acceptance of trans people as the gender they identify as. This movement is unprecedented. Never before have we seen such widespread support of trans people in protests and in mainstream culture. We are seeing a massive shift in language while we are simultaneously questioning our understanding of gender conceptually speaking. Give it maybe a decade at most, and it is highly probably that will see dictionary definitions like “adult human female” be changed to reflect the reality of what gender descriptively is in the social context: expression, and identity. We’ve already seen a shift in language for definitions like “trans woman” and “trans man” so it doesn’t appear as if it will be much longer before the dictionary definitions don’t reflect the argument you’ve made thus far.

TLDR: Gender is partly expression and partly identity. We see this in the historical and socio-cultural usage of gender as a means to oppress females, but also a cultural phenomenon and means of social identification. That being said, we are seeing a upward trend in support of this idea, and affirmation of transgender people as the gender they identify as (and therefore are).

It's feminism's aim to rid our culture of gender

Absolutely, it certainly is. Therefore, we should eliminate the social demand to adopt the social role of one gender or another, and accept transgender people as their gender, as their acceptance is one of the primary steps we can take to eradicate gendered norms, expectations, and demands.

You claim that in such a post-gender world gender would still exist as some nebulous, non-specific, undefinable entity.

A postgenderist society would not come into fruition until gender identity is also abandoned due to its unpragmatic nature, which can only occur with the eradication of societal demands of conformity to certain gender expression. That being said, the most efficient manner to eliminate mandated gender expression is to accept transgender people, as their accept directly implies that there is no biological or societal objective truth to gender. It is simply a nonsensical idea used to oppress females, but by accepting trans people, we are removing that motive that is induced by gender in its current state. With that all being said, I’d much rather live in a purely gender-identity-based society where people have the personal freedom to pursue the gender they most identify with rather than one than forces conformity based on arbitrary expression, most prevalent in harming females.