Might give some insight to support here if my hunch is right. Otherwise I'll get my roommate to DL this and do it.
If you are running LoL on a PC (preferably windows 10, or 7) go to C:\Riot Games\League of Legends\RADS\solutions\lol_game_client_sln\releases
There should be a folder in there with some version number. If there is more than one then try the highest version number one I guess. Go there, then cdouble-click the 'deploy' folder and then double click the "League of Legends" program/icon you see.
This should be the file that actually runs a game. I'm curious what happens if you just run it by itself (like this and not having the launcher run it). Im still having the same popup error shown below when I do that. that is the same error I get running the game normally though and I'm wondering if the LoL client is somehow not passing relevant information to this file when it tries to run it, hence my error. Its a little strange that I see the same error message when trying to start this by itself.
I'd appreciate it if anyone could just try double clicking that program and seeing what happens.