And you could absolutely do that. We don't have GRCs in the US. A person's transgender status is based entirely on informal self-declaration. And, as I'm sure you're all aware, thanks to the antics of Philip/Pippa "I'm a Woman on Wednesdays" Bunce, trans people don't have to consistently ID as the opposite sex. They can arbitrarily decide to be men or women at any time, depending on which is more convenient at the moment.
So, there is absolutely nothing stopping female police officers from identifying as men when a trans-identified man wants to be searched by a woman, and then immediately going back to identifying as female once the suspect has been booked and processed into the system.
In fact, there is nothing stopping ANY American woman from identifying as a man whenever it's convenient to do so, and then immediately going back to IDing as a woman when she feels like it. THERE ARE LITERALLY NO RULES.