Court order to change the name on the BC, and depending on the birth state law a court order or surgery letter to change the gender on the BC. At least, that is what I went thru living in a state other than my birth state. In my case, I got a court order for name and gender change, but my birth state required surgeon letter to change gender on BC, so I sent court order and surgeon letter to change name and gender on my BC.
I will add that my State Driver's License was able to be changed for name and gender with just the court order.

(...same with passport, SSN, and EVERYTHING else. Stupid BC HAD to have the surgeon letter...)