I don’t know for sure but I can think of 3 reasons why it would not work. It may not fit in the space in the 6s (it’s pretty tight in there) It may not connect electronically. Each camera model is made for a specific model of iPhone. The 6S may not be able to work the camera (focus and OIC).
It sounds like your phone is in Low Power mode. Got to Settings-Battery and there should be a slider to turn it off. I believe the phone will go into Low Power mode automatically if the charge is very low. Hope this helps. There may also be a problem with the battery.
Check: Settings-Battery-Battery health. If it’s below 90% (my 6s was at 88%) you may be able to get a free battery replacement from Apple. I did mine a few months ago and it’s like having a new phone. Check this site: https://www.apple.com/support/iphone6s-unexpectedshutdown/
It sure sounds like the LCD or the backlight. You'd surely want to check the connectors. Before you open it you could try to GENTLY squeeze the case edge all the way around the perimeter. If it changes it may indicate the cause. When you turn it on do you see any change in the screen? If it's dead black then it could be the backlight. If there's some change (less black) it could be the LCD. Hopefully it's just a loose cable and opening it and reseating will fix it. Good luck.