Note: I have found no solution to use the PC as a bluetooth game pad. This is all workarounds.
Streaming games from Android to PC
If you have an android device, and you have a game installed on the android device, and you want to control it from the PC... you can use one of many streaming solutions.
Ignoring those that require special hardware, and looking for one with gamepad support leaves us with almost nothing...
However, there is one that can do a workaround: Wondershare MobileGo. Now, it will not present itself as a gamepad to your android device... instead it will allow you to create touch input for a virtual (on screen) gamepad. You can configure the positions for touch input
Ok, ern... it takes keyboard input, not gamepad input...
I have two alternatives to take gamepad input and turn it into keyboard input for your PC:
Emulating Android on your PC
If you want to play an Android game, you could be able to emulate Android on your PC and play there.
I have a few alternatives for doing that:
Remember that you can use the solutions from the section above to use your gamepad as keyboard.
Streaming games from Xbox to PC
You can use the Xbox app for Windows 10. If you do not have the Xbox app installed, you can find the Xbox app on the Microsoft Store.
The Xbox app has a stream function that will connect to your Xbox One or Xbox 360 on the LAN, and allow you to play on the PC with whatever input devices you have. You will find it on the console icon, second from the bottom on the bar on the left.
I presume you do not want the video output on the PC monitor, hopefully you can connect the desired video output device to your PC, hopefully you do not need adapter hardware (hdmi is pretty common these days).