Wow, this one was in my wheelhouse, in 1994.
On a morning in December 1993, my legal name change was approved at my county courthouse. After stopping at the local Bureau of Motor Vehicles (Ohio) to get an updated driver's license with "F" in the appropriate spot, I proceeded to the FAA's local Flight Standards District Office. I presented a copy of the court order, and was issued a new pilot certificate, again with the new name and "F" in the appropriate spot.
On cloud nine, I went home, changed into my boy-drag uniform and went to work (I think my report time that day was 1:30 pm). When I got to the airport I did a quick stop at the ID office, where I was issued an airport identification badge with my new name. After I checked in I went by the Chief Pilot's office, made copies of my new license, and asked if they wanted me to sign legal documentation, like dispatch releases and such, using my old name or new. After hemming and hawing for a few, the assembled braintrust decided new.
I flew all my trips for two months in boy-drag, signing paperwork as Captain Diane, while the airline tried to figure out what to do about me. During that time I visited my local aeromedical examiner, who issued me a new Medical Certificate that said Diane AND Female. No questions asked! On February 2, 1994 I flew my first trip as a female captain with long, red finger nails. RLE had begun, and none of my first officers objected to being trapped in the pointy end of high-speed aluminum tubing for extended periods of time with me.
I was not the first airline pilot to transition, and I certainly wasn't the last. Back in 1994 one of the really big legacy carriers had several transsexual pilots, but I was the first at my airline. It took the company about nine months, plus a year of haggling, to put me into a position where resignation was my only logical alternative, but it was great time while it lasted.
That's a long story to say apparently the FAA's attitude has deteriorated over the last 20 years. They started out so good on the issue. I'm sorry that Jessica has to go through bureaucratic BS.