I would suggest trying to get a game-pad with numerous buttons (i.e. 12 to 20) and an integrated mouse such that you can access all of them using a single hand simultaneously. I know that they make them for gaming (and other functions), but I forget their actual name.
You may be able to find it if you either search for it or ask someone more informed than I am.
You should be able to bind the buttons on the game-pad to your relevant keys (QWERDF, Space, Tab, etc.).
There should be both left-handed and right-handed versions, though you may need to have it shipped.
I don't think the game lets you adjust mouse-buttons through the interface, but you may be able to do so by editing the PersistedSettings.json file. It's not exactly the same thing, but you can get a rough idea of how to do so here:
https://www.reddit.com/r/leagueoflegends/comments/3vgw7a/how_to_bind_attack_move_to_left_click_with_full/