Whether it is logically inconsistent depends on how you define several terms. Of course everyone is going to choose definitions that make their points of view self-consistent... so that obstacle needs to be cleared first. You very quickly fall down a relativist hole.
The best way out of that hole is to describe things literally, with as little interpretation as possible.
For example, if you interview a hypothetical person Alex, and Alex claims to be a feminist, then you could write: "Alex, a self-described feminist, ..." but should perhaps avoid writing "Alex, a feminist, ..."
The first only relies on you, the audience, and Alex agreeing on the very basics of the English language. The second implies some sort of agreement between you and Alex on what it is to be a feminist; basically you are injecting your assessment of Alex's beliefs. In the first, the audience may disagree with Alex, but they should have very little room to disagree with you. In the second, the audience disagreeing with Alex pulls you into the fray as well.
You will find irrational extremists on either side of every political issue. In the case of gender relations, you have the religious right and MRA wackos on one extreme and the mysandric, transphobic "RadFem" fruitcakes on the other.
Much of "radical feminism" holds that gender isn't about expression, and that you can't switch genders as much as you can switch races. This makes people extremely uncomfortable, of course
Do you agree with the following definitions commonly used when discussing gender issues?
Sex: Assigned biological category -- male or female: designated at birth by visual assessment of anatomy based upon presumption of reproductive role.
Sexual Orientation: Term for an individual's physical and/or emotional attraction relative to their own sex such as lesbian, gay, bisexual, or straight.
Gender: The social meaning given to biological sex.
Gender Expression: External characteristics and behaviors associated with gender that are socially defined and associated with masculine or feminine. For transgender people, their gender expression doesn't match their biological sex.
Transgender: People who identify with a gender that is different from their biologically assigned gender.
Intersexed: Describes people who are born with external genitalia, chromosomes, or internal reproductive systems that are not traditionally associated with either a "standard" male or female.
Cis-gender: People who identify with the sex or gender they were assigned at birth.
Gender is generally assumed to be defined as that part of 'sexual identity' which is a social construct and hence changeable.
It seems logically inconsistent.