In your game controller VI, you may call Open VI Reference, but you may not call front panel methods or properties. What you are trying to do is possible. Here's how:
Use the Web Publishing Tool to make an .htm file for each VI that you want to display dynamically. You will have an .htm file for your game controller VI and one for each game - checkers, chess, backgammon, etc. Make sure you've placed all the htm files in RT's web root and all your game VIs somewhere on the RT disk.
When you launch a game, the game controller must call Open VI Reference followed by the VI Run method, which will place the game in memory and run it. Then all you need to do is load the game's web page.
If your RT box is given VI Server access to your host, the game controller could talk to a VI back on the host. The VI on the host could do anything - launch a new web browser with the game's URL, or refresh the existing web browser with the game URL, etc.