Since the League of Legends client is web-based, our Desktop app's game capture won't be able to detect it. The Player Desktop app hooks directly into games that run on DirectX 9-11 and OpenGL.

You'll need to use streaming software with screen/window capture like XSplit Broadcaster or OBS Studio to be able to stream the lobby screen since it works separately:
https://www.xsplit.com/broadcaster/faq/faq-league-of-legends