The woman shall not wear that which pertaineth unto a man, neither shall a man put on a woman's garment: for all that do so are abomination unto the LORD thy God.
God commands males to wear men's clothing and females to wear women's clothing. Anyone who does otherwise is an abomination to God.
Although the Bible doesn't specifically address the transgender bathroom issue, it seems to say that transgender people are an abomination to God.
So maybe the Biblically Correct solution is to have three separate bathrooms: one for males, one for females, and one for abominations.