I have a HP display and to switch video inputs I press some button on the display twice.

For the keyboard and mouse I use some USB switch.

But since you are going with Synergy you may not need that second part. I wonder why you don't use some remote desktop software.

Ok, SPLIT SCREEN is the clue that you might want to have both computers on screen at the same time. The answer is NO. No that will not happen without some expensive hardware that costs far more than a second display.