The reason that this sounds like a bigoted solution is that there aren't enough trans athletes in the country for a league of our own to have actual competition, and just doing the awards separately obviously isn't enough since the bigots are still complaining that we are allowed to participate in sports at all.

Equality will always feel like oppression to those who benefit from inequality, and there's no liberal workaround solution to the objections from the haters because their claims about "fairness in sports" are insincere. If the point of the divide between genders is to make space for women in sports, then excluding some women because of personal characteristics out of their control is unfair. And if the argument is about some (real or imagined) performance difference between cis and trans athletes, then the fair thing would be to set up ranked leagues so that everyone can compete with people near their own level of ability.

Sports are about friendly competition, healthy opportunities to better ourselves by testing our bodies and our skills against our peers. The right thing to do here is pretty obvious, the problem is that there are people who don't want to accept trans folks as a valid part of society and we won't be fixing that by fiddling in the margins of sports rulebooks.