Thus it is unquestionable that there can be a gender to the mind.
No, it is unquestionable that there can be a
gender identity to the mind. Gender and gender identity are not the same concept. An awful lot of trans ideology's arguments rely on equivocating them.
What is there to distinguish between them?
"Gender refers to the characteristics of women, men, girls and boys that are socially constructed. This includes norms, behaviours and roles associated with being a woman, man, girl or boy, as well as relationships with each other. As a social construct, gender varies from society to society and can change over time.
...
Gender and sex are related to but different from gender identity. Gender identity refers to a person’s deeply felt, internal and individual experience of gender, which may or may not correspond to the person’s physiology or designated sex at birth".
- World Health Organization
To oversimplify: your gender identity is the sex-linked category you think of yourself as being in; your gender is the sex-linked category
other people typically think of you as being in. That's what "socially constructed" involves.