I know you say that you are sure you're using the right account and password, but my guess is still that maybe you aren't?
The user info you use to log into https://t3aonline.net/ is [potentially] different from your server log in info.
So log into https://t3aonline.net/, then go to the accounts page (https://t3aonline.net/account/) and verify you are using the server log in name and password to try and get into online on ROTWK.
Otherwise to verify that you're doing the "run as admin" part correctly, make sure you're:
1) find your lotrbfme2ep1.exe file in the RotWK install folder.
2) right click it and set the properties to always "run as admin" in the compatibility tab
3) open up t3a launcher (might need to right click and run as admin)
4) click the Rotwk button
Hopefully one of those things is the problem..?