Rather than splitting by gender it should be split by sex, or birth sex, if you prefer.

The problem is that doesn't entirely work for trans women. Hormone therapy during their transition will reduce their performance to the point where they mostly can't be competitive with cis men anymore, while they'll still dominate against cis women.

Everyone's relative chromosome string should be determined, like XXY and that should be the category, devoid of any loaded words like "man" or "woman". If you're XY, you don't get to got against XX athletes, regardless of the identity of anyone involved.