Mirroring display in Recovery Mode with faulty lid sensor and LCD?
The solution that worked for me, and was suggested to me by someone over on Reddit, is to remove the bottom of the Macbook and unhook the LCD cable. Once I did that, the external screen acted just like the internal screen. I could see the Apple logo, the recovery mode welcome screen and all that jazz.
There's teardown instructions over on iFixit, but I didn't need to follow all the steps. I just needed to undo the screws (steps 1-8) and unhook the trackpad cable (steps 9-11) to get the case open far enough to unhook the LCD cable (steps 19-23)