Your friend originally wrote "Here are my thoughts, in no particular order", and I think it would be helpful to help her refocus by extracting one or more concrete position statements or synthesizing one common theme to put some "handles" on this slippery discussion, and maybe discuss them one at a time. Start by putting names to what you disagree with. Is it any of:

• "biological sex is binary, not bimodal"
• "race and gender relations are not analogues"
• "sex is the single most determinant category for athletic performance"
• etc.

Ultimately your friend seems to be summing up her disagreement with a very general, unspecified thought:

I strongly disagree with what I think you’re suggesting about sex- and gender-based policies. I say this on the basis that rejecting trans-women and men works to support patriarchal structures that are universally harmful [...]

However, I assume what you're suggesting is not "rejecting trans-women and men" based on patriarchal structures, or even rejecting them at all, but having policies based on what they physically are instead of what they feel or wish to be. It might help to try and specify for her what you actually think these policies are, and what determinant of fairness they should be based on.

I just wanted to add, the idea of a dichotomous "blackness"/"whiteness" is ironically much more of a spectrum than the arguably binary gender/sex argument. (Or bimodal or whatever: such splitting of hairs really does not matter to the greater point.)

People can much more obviously be 1/2, 1/4, 1/16 or some other mix of African+European ancestry (with honourable mention for the other continents) ultimately making race meaningless. The same will never be true of physical sex.