I'm a PS3 & PC player. I played on PS3 for a while then moved to PC, but on the same television. I read all of your solutions and none of them seemed to mention any form of DPI settings.

Forewords:
I'm sure depending on the generation of your TV, it may or may not operate more like a large tablet PC (Smart TV), meaning that it more than likely has some kind of GPU-like device in it to display the UI.

Background:
During my use of PC, I ran into an issue regarding the DPI-Scaling options where I had to disable them and run my screen in 720 in order to fix full-screen displays of certain games and applications, to make them work properly.

Suggestion:
Look on your TV for any settings regarding DPI-Scaling and see if you can disable or enable it, or even modify it. Some UI's may allow you to literally SCALE what is viewed on the screen. For instance, you mention that you tried to mess around with the edges by moving the display itself, but this would lead the opposite edge to be cut-off more, and so on. DPI-Scaling is something some console games allow you to do upon initial play of the game.

Typically what you get is 4 arrows on each edge of the screen, and you use your + and - (vertically and horizontally, respectively) to stretch or unstretch the display to fit your display properly. Now because of this, I'd recommend performing the action in 720, although your console may automatically set the display to 720p since this is the limitation of the game, but just to be safe try it.

I hope this helps if you can find a setting like that, again look for references in your settings to the word "DPI-Scaling" or "Dots Per Inch", etc.