I've been running on a bug with the Riot client itself.
Having the same logins for both Live and PBE Servers bugs out the Client.
If I try to connect to PBE Servers, it will automatically switch me to Live EUW Servers (As you can see on the video bellow I'm French so this is my main server, don't worry about Random picked up server).
Whatever I try to open the Main or PBE client, it switches me anyway and signing-out is not working either to help me connect to my PBE Account.
I'm pretty sure this is linked from me using the exact same creditentials for both servers. So I'm probably not the only guy running this issue then.
Here is the video I captured minutes ago (after your post)
I'll try in the meantime some basic Troubleshooting stuff (Repearing, Reinstalling, ect) and eventually have different logins.
Gl hf !
EDIT : Changed the logins and I learned something : using your Live account logins will connect you to the Live Server anyway! Whatever you picked PBE or Live (EUW).
Launching the PBE client and selecting EUW Servers and entering my new logins of my PBE account connects me to PBE.
So the Selecting Region button has literally no impact on the Riot Client.