hi jagfel!
I just changed my Facebook about a month ago, (first name only, kept my same last name) and had no problems at all. It was pretty simple and straightforward, Edward to Erin. If your name is less common then that also may have raised some sort of flag. I had also changed my gender marker on there about a year earlier, even though it wasn't made public. (It did post things like "Edward changed HER profile picture."). So if you're changing from one heavily-gendered name to another, it might be helpful to change your gender marker first. That's totally just a guess, though. Good luck both on the Facebook change and in getting your ID docs.
Erin