Well, if it is the Logitech UniFYING receiver, then that wouldn't work. That's made for Logitech products and they haven't made what you're looking for.
Well, if it clips onto your screen, why would it need to be bluetooth? I don't think there is one like that. Just a wireless mic to clip onto your shirt would be cool, but you'd have to charge it pretty frequently.
If you are talking about The Logitech Unifying reciever, that is not "universal" and it's not a bluetooth device although it uses the same band. If it's attached to your screen why not just use a wired microphone? You're not moving it around.
Well, if it is the Logitech UniFYING receiver, then that wouldn't work. That's made for Logitech products and they haven't made what you're looking for.