The GPU seems not so powerful for the level of optimization LoL has
That makes no sense. First of all, the GPU is absolutely overkill for LoL. Secondly, the more optimized something is, the less processing power you need.
OP, what are you using monitor capture for? As it's usually incredibly inefficient, try disabling it to see if it improves the performance.
Also, not related to your FPS but useful otherwise, it seems it takes a long time for you to connect to the Twitch server. It might be worth it trying out a different server.