Instead of trying to rename the Koch Theater, why don't we just give an additional name to the actual stage: The George Balanchine Stage at the David H. Koch Theater at the Lincoln Center for the Performing Arts. Thereafter, those who object to the theater's name can just refer to the facility as the Balanchine Stage. Or how about the Balanchine Proscenium at the David H. etc., etc.
Thinking about this further during lunch:
The George Balanchine Stage would be so-named following the waiver of fees from The George Balanchine Trust for three years. That would amount to about $1 million - a nice reduction in expenses, a nice gift. In the orchestra pit, there would be the Stravinsky Podium which would be funded by gifts from elsewhere.