LogFAQs > #970830627

Wow, no one's mentioned the real reason -- there are no "male leagues" in a lot of sports, there's the General leagues, and the female leagues. Confusingly some sports do call these "Men professional leagues" or something similar, but often there's no rules about who can compete, men or woman, or anyone else can.
Therefore, If Men or Women can compete, neither-side of the belief spectrum would have any issue with a Transman competing, no matter what their beliefs.

The General leagues anyone can compete in -- and actually, here's some links about woman competing, and/or winning in the general leagues of sports:

https://en.wikipedia.org/wiki/Kelly_Kulick -- Kulick won a major PBA Tour tournament.

https://en.wikipedia.org/wiki/Ila_Borders -- The first female pitcher to start and win a men's professional baseball game.

https://en.m.wikipedia.org/wiki/Manon_Rh%C3%A9aume -- The first woman to play in exhibition game in any of the major North American pro-sports leagues.

https://en.wikipedia.org/wiki/Erica_Enders -- Enders won the NHRA Pro Stock World Championship for a second straight year in 2015.