Every law needs a purpose.
The reason laws that allow transgender people access to facilities are being proposed due to the harassment transgender people receive in the public square.
Seventy percent of transgender people have been assaulted, harassed, or kicked out of facilities when trying to go bathroom alone. http://williamsinstitute.law.ucla.edu/wp-content/uploads/Herman-Gendered-Restrooms-and-Minority-Stress-June-2013.pdf
Frequently this harassment occurs through the use of the authorities. People don't often have guts enough to be a bigot to someone's face, so they will call managers or other authorities to do it and have transgender people hassled, harassed, and removed. Police are exceptionally unkind to transgender people. They can be outright hostile. A law that requires someone to show ID is no for transfolks, in fact, it makes life worse.
As for harassing and committing crimes? If a man, who looks like a man, goes into the ladies room. He's going to attract attention. As such, if it gets to the step of asking him about about his gender identity, he's failed the first step in committing a crime. He attracted attention and was memorable. It would be easier for him grab a mop and sneak in claiming to be a janitor. No one notices janitors.
If someone is being a creeper in the ladies room, they'll be kicked out for being a creeper. Creeper is not a protected class. Claiming to be transgender is never going to be a free pass for harassing behavior.
Besides, women can harass other women, there's nothing special about gender. I'm a member of a women's gym - and they ban cell phones in the locker rooms.
The things you are describing are not things that transgender people do. If the society evolved to the point that gender presentation did not matter in any way and could not be questioned, we'll end up with unisex bathrooms everywhere and it's a non-issue.
I'll leave you with massachusetts law on gender identity, this would only apply if things got as far as court:
'Gender identity'' shall mean a person's gender-related identity, appearance or behavior, whether or not that gender-related identity, appearance or behavior is different from that traditionally associated with the person's physiology or assigned sex at birth. Gender-related identity may be shown by providing evidence including, but not limited to, medical history, care or treatment of the gender-related identity, consistent and uniform assertion of the gender-related identity or any other evidence that the gender-related identity is sincerely held as part of a person's core identity; provided, however, that gender-related identity shall not be asserted for any improper purpose.
The last sentence basically covers all the problems you have described.