I suspect the driver for the headphones are not compatible and the Sony website says there is no drivers. I don't have a bluetooth headset so I am flying blind here.
1. Go back to services.msc be sure both 'Bluetooth Handsfree Services' and 'Bluetooth Support Service' are set to run automatically. Go to run and open devmgmt.msc and uninstall the bluetooth driver. Now go to run and open regedit and navigate to...
With CurrentVersion selected over on the right pane look for (ab string) CurrentVersion. Right-click CurrentVersion and choose modify. Change the value from 6.3 to 6.2 Close out all the files on the right pane and close the registry. Reboot the system and reinstall the bluetooth headset.
If that still doesn't fix the problem then there could be unfroseen damage to files. Reverse what you did in the registry above back to 6.3 Once that is done go to run and type cmd right-click runas admin and type sfc /scannow. Be sure to do Microsoft updates as well. I am speculating that the headphones worked in Windows 8, but now not working with 8.1 I am surprised Sony doesn't have a solution to this problem. Have you thought about upgrading to Windows 10? I thought I would never hear myself suggest that. lol However, I am speculating that Windows 10 would have to have much better Bluetooth support because Windows 10 according to Microsoft will run on anything or has plans to have it run on smaller hand held devices. You can't accomplish that without having better support for sdcard/micro-sd, usb/micro-usb and blueooth/wifi etc. It would be a coin toss, but you would have the latest OS for free and can reverse it if need be. It seems you have gotten a hardware device that got caught between the phase of many OS upgrades. You may ultimately have to buy a new headset.