Chosen Solution
I have an iPhone 6s Plus with a on and off working home button, I have replaced the home button and it does the same thing, it will work for a while and then it just stops working, just to work again later on. One of flex cable connectors at the top of the screen (specifically one for the home button) is slightly split. Would this be the cause?
Yes, replace the screen and it should fix your problem. If you have the factory home button I would put it back in so touch Id will work.