there may or may not be a right to compete, we dont need to debate that because the overwhelming fact is that noone has a right to be 'competitive'; being competitive is a mixture of genetics, assistance and hard work.
I compete, but i am not competitive, since my received genetics are poor (my uncle was a competitve xc runner, so the genes are somewhere in the fam, just not expressed in me).
same with transgenders. when they choose to transition, they choose to go into a different pool. this does not alter their right to compete, it changes their relative competitiveness; for f-m it make them uncompetitive, from m-f it makes them unfairly competitive.
and thus we have to protect women's sports. trans are not excluded from competing.
you havent mentioned in your 'solution' what changing room they use.