If I am understanding you well...
You have in a way, two source content to be connected to your projector?
This "Switch-Box" allows you to connect up to 4 source content to any HDMI display or likes. One at a time!
Perhaps we can start with the original content source connected to your surround system if it supports -Pass-Through- then onwards to your projector? In which case, you would not 'need' this "Switch-Box".
For what you wish to accomplish, seems a signal splitter or "Y" adapter, if is made for HDMI, is what you may need.