This issue seems to be common to 2015 models. Re-installing the OS in-place is unlikely to fix this issue now that macOS uses a signed & sealed system volume.
Bluetooth configuration files are stored here:
/Library/Preferences/SystemConfiguration/com.apple.Bluetooth.plist
/Library/Preferences/com.apple.Bluetooth.plist
/Users/*/Library/Preferences/ByHost/com.apple.Bluetooth.*.plist
/Users/*/Library/Containers/com.apple.preferences.sharing.SharingBluetoothService
You could try deleting all of these to reset the Bluetooth settings, but I suspect it's an OS bug that will require an update from Apple.
To that end, the best option for fixing that is for us all to submit a bug report to apple, and reference this thread. The more people that do this, the more likely it will be fixed.
https://www.apple.com/feedback/
(click "MacBook Pro" or "MacBook Air" under "Hardware").