Sorry I didn't realize the problem you faced with the log out option. Although I'm not quite sure that I understand why a developer would choose to include clay.io in their game, only to auto log out all of their users (perhaps I'm missing something).
Anyway, your solution of making a method to call a log out window sounds absolutely fine. Could it also include a link to delete an account if the user wishes?
I think both these options would make users feel more in control of their choices. But don't worry if the second point is too complicated. I can just add some text to inform users they'll need to visit the website to delete their account.
Thanks again for your time in this matter, it's greatly appreciated.