I'm not too familiar with streaming and what encoding is available out there or where the stream gets "processed" (cpu or gpu or both)
this being a desktop system and if you're playing 1920x1080, a 460 should have no problem running the game all maxed
If you think your rig is not powerful enough for streaming, check your utilizations first. I really doubt you're topping out on CPU or GPU.
I am more inclined to think that streaming is limited by your upload rate. You need to find out what your upload actually is when you're streaming (get something like bandwidth monitor) and also consider what is the settings for the stream? Is it trying to send your full resolution out and struggling for bandwidth? Or is it trying to send raw stream with zero encoding on it?
btw, as a fellow league player, hope you get this working there's really very little video resources for league out there, seeing how riot still cannot work out a replay feature.