How can I unpair a Bluetooth device from a recycled computer?

How can you unpair a mouse from (old and recycled) Mac computer A so that you can pair it with (new to you) Mac computer B?  

Although the mouse was on the Device list of computer A, you can no longer use that computer, so you cannot unpair the mouse using Computer A.  Since the mouse is not on the Device list of computer B, you cannot use System > Bluetooth to unpair the mouse using Computer B.

Since the mouse was paired to computer A (and was not unpaired from it), the mouse won't respond to the "seeking to pair" signals from computer B.  In this situation, is there any way that you can reset the mouse to being unpaired or do you have to discard it?