Grade A failsauce right there. Let's start at your first assertion. Multiple reports indicate that no transgender individual has ever been arrested for sexual misconduct in a public bathroom, while at least three Republican lawmakers have.
Former U.S. Senator Larry Craig, Florida state Representative Bob Allen, and Mississippi Congressman Jon Hinson were all Republican lawmakers, and were all arrested for sexual misconduct in a public bathroom. But wait, there's more!
SC GOP Rep. Josiah Magnuson attacked and harassed fellow Republican
  NV Sheriff Gerald Antinoro - sexual harassment/assault
CA Assemblyman/gubernatorial candidate Travis Allen - sexual harassment
  Knox County Judge Steve Sword, appointed by GOP Gov Haslet, sentenced man convicted of 9 counts for raping his 14yo daughter over 2 years to just 12 years rather than the recommended 72 because of his Christianity
  Mississippi Republican State Rep. Doug McLeod snapped "as he often does" and punched his wife in the face for undressing too slowly
  Jonathan Parker - former Idaho GOP Chair - arrested for stalking, masturbating outside apartments
  David Narramore, KY GOP State Executive Committee and Letcher Cty Chair, arrested for exposing himself
And why do you even care what someone has between their legs? What is it with this sick fascination with our genitals? And it is sick and not the good kind of fascination, I've seen the good kind. It's become a pathological and perverse obsession to know what tool kit I'm working with, what position I use under the hood and with who's hood am I under. Ewwwe! It's creepy af, like sexual predator kind of creepy. How about you mind your own business and I'll mind mine? You've got enough on your hands just doing you, don't worry about me.
"All . . . will bear in mind this sacred principle, that though the will of the majority is in all cases to prevail, that will to be rightful must be reasonable; that the minority possess their equal rights, which equal law must protect and to violate would be oppression."
-Thomas Jefferson
Majority rule is limited by the supreme law of the constitution, which protects the rights of individuals from the majority idiocracy.