the whole subject is so politically charged
The only thing 'politically charged' is the fuss male-bodied and male-socialised people are making about their demands. Male fuss is political, serious, meaningful and important. Female fuss is silly, little, domestic hysteria. 
whatever physical advantages transgender women have – and they certainly exist – they are not nearly as large as the sociological disadvantages.
sport must strive to be as inclusive as possible
Sociological disadvantages are what stand in the way of inclusivity in sport.
Physical advantages are what sport is about.
So, in order to include trans-identifying people in the opposite sex sport, perhaps a handicap system should be used.
Transwomen players could carry extra handicap points (like the extra weights added to a jockey who weighs too little), and transmen players could be given head-start points (like the advantage a weaker golfer gets given when playing against a much stronger golfer). In ball sports the transwoman playing against a women's team could have to play one-handed, whereas the men's team fielding a transman could be allowed an extra transman player to even out the differences between male and female bodies. Physical advantages and inclusivity both dealt with.
(Disclaimer: tongue firmly in cheek. TW do not belong in female sports.)