I am in the same boat as you. I know you posted this question two years ago so have you found a television that has this capability? If so please let me know. Upon all of my research all I could find that Vizio used to do pip, but stopped last year.
I went to Best Buy and I asked one of the TV guys what TV would allow me to play games and watch TV at the same time, basically PIP side by side split screen. He said none of them did that anymore and left me alone. I did not believe him so I come where the experts are to help me with the question. Is what I want possible at all? If so which TV should I get?