I have had that issue off and on with pairing my iPhone6+ since iOS9 but got much worse in iOS10. Funny thing is when my thumbprint reader works and apps are more buggy then my bluetooth pairing/functions are terrible. When my thumbprint reader is nearly useless and my apps work with minimal issues then my bluetooth seems to function better.. hasn't been a happy medium since iOS9 which was more often the bluetooth was working situation.
Fresh load of 10.11 El Capitan and my MacBook Pro (15" mid 2012) ran flawless with no issues and being a tech I did extensive testing of all functions (2.3GHz i7, 16GB RAM, SSD). I have a bluetooth MS mouse, and bluetooth Phillips headset that worked very well. I updated to Sierra 10.12 and began having issues with both my mouse and headset. Additionally my internet access has been sluggish or very buggy. Reading up on common Sierra issues I decided to try a fresh install to eliminate the issue being caused by an upgrade install. Many performance/glitches and app issues were resolved including boot up & shutdown stopped having hang pauses. It however did not do much for the performance of my wifi or bluetooth.
I am working on investigating the drivers and service functions of Sierra to see if I can find the source of the issues. Maybe another tech already found a solution but I have not stumbled on to it yet. Good luck in solving this and hopefully Apple will release a fix for Sierra and iOS to resolve these problems...