Yes, you may play the game on closed source software while the game is distributed as a module for that software and the game still remains clearly open source under the GPL and the terms of the GPL are met. This is no different from using GPL files on for example Microsoft Windows (closed software) inside Adobe Photoshop or Notepad (closed software).
As a software and/or game developer you can’t include the game or parts of it into your project using other licenses. The GPL of the game can’t ever be removed and the game can’t be re-licensed by you. What you may do however is to enable your software so that the game is playable on it, using the GPL.