In my opinion, the radical feminist VS gender identity debate boils down to materialism vs dualism and feminism vs post modernism.
To explain:
Mind–body dualism, in essence, postulates that human beings have souls. It's a philosophy that denotes either the view that the mental is non-physical, or that the mind and body are distinct and separable. Therefore, one can be born male and have a female soul/spirit/essence/psyche/personality/identity. It is compatible with many religions and was promoted by famous philosophers and theologians (often both.) It is a form of spirituality.
On the other hand we have materialism. Radical feminism, marxism, socialism and communism come from materialism. Materialism can also be found in science, such as the theory of evolution.
Materialism is a form of philosophy which holds that all things, including mental states and consciousness, are results of material interactions of material things. Mind and consciousness are caused by physical processes in the body without which they cannot exist. It opposes spirituality and dualism and negates the concept of a soul. Therefore, there is no female soul inside of a man's body because there is no soul at all, female or male, only a body. That man's body is male because of its sex which means his personality and thoughts are that of a male.
The other conflict is between feminism and post modernism.
Post modernism, as a philosophy, postulates that there is no "truth" because everything is relative. There is no objective reality, just opinions, perspectives and thoughts. Nothing is real, because everybody is different. There are no facts, only viewpoints. Therefore, postmodern philosophers spend a lot of time discussing theory and abstract concepts instead of the material reality of class struggle and its consequences. Think of all the discussions about "identities", "what is a woman" and how to "feel empowered." In this world, sexism is something relative and abstract, something to talk about, not to fight. To quote Catharine MacKinnon "Their critically-minded students are taught that nothing is real, that disengagement is smart (not to mention careerpromoting), that politics is pantomime and ventriloquism, that reality is a text (reading is safer than acting any day), that creative misreading is resistance (you feel so radical and comfortably marginal), that nothing can be changed (you can only amuse yourself)." Post modernism is directly linked to individualism (a social theory favouring freedom of action for individuals over collective) and neo-liberalism (there is no public interest, only individuals' interests) positioning itself against collectivism, class consciousness and class struggle, all core tenets of feminism.
Feminism is a political and materialistic movement that seeks to liberate women (radical feminism) or at the very least to improve the very real, very concrete, material lives of some women (liberal feminism). It postulates that men and women are social classes and that men are waging a war against women, and have been for centuries. The oppressors and the oppressed are clearly defined, it's not "relative", it's not a viewpoint, it's not "personal": it's historical, political and factual. Strategies and actions are used to free women from male supremacy (radical feminism) or to make said male supremacy more bearable (liberal feminism). In this movement, the idea of Woman the abstract concept or Woman the spirit/personality/soul in a body makes no sense and is considered misogynistic.
To summarize:
-> Trans activists believe that physical bodies are irrelevant or, at the very least, separate from the mind, that reality changes depending on how we think about it, and that women are a personality (and personalities are abstract concepts).
-> Radical feminists believe that women form a social class that is oppressed (even enslaved) by men precisely because of our physical bodies.
Those two positions are irreconcilable.
