Does windowed mode cause a performance hit?
Not to my knowledge, but my thinking is that perhaps the game, with its many faults on PC, is trying to full screen across both monitors, failing to do so, then perhaps putting the second monitor into standby mode. When I've had a HDTV in the past as a second monitor, it would be triggered into standby mode by the craziest things, even by alt+tabbing. I currently have a projector as my second 'screen' and it does not like displaying different content to monitor #1. When the screens are duplicated it's all fine and dandy.
It's a bit of a stretch, but might be worth a try in Windowed mode. If the TV is immediately turning off when Resident Evil 7 is booting up that tells me that the monitor isn't getting a signal, likely because the game is in dire need of some bug fixing and optimisation.