I'm curious - keep us updated with how it goes. I think it is worth trying a dumb single link adapter (available on Amazon for peanuts) and seeing if the concept will work, because the Lightning-to-HDMI adapter might not work well with a secondary video conversion since the adapter already performs
some trickery:
"...the iDevice is actually sending an H.264-encoded image through the Lightning interface to the adapter, where a small ARM chip decodes that information and outputs it over HDMI."