Kensington PocketMouse disconnects but Bluetooth still registers it as "connected" and can't disconnect or reconnect

This seems to happen more often when my computer goes to sleep before the connection with the mouse is dropped due to inactivity. The mouse is the only Bluetooth device I use with this computer (which it also has problems with "discovering" the device automatically, but it's a few clicks to get it to work, so it's not a huge deal).

My mouse (a Kensington PocketMouse/PilotMouse - it's quite a few years old) will disconnect/go to sleep, but the Bluetooth on my computer won't register it as having been disconnected. I can't manually disconnect the mouse from the menu, it won't allow me to turn off the Bluetooth system, I can't remove the mouse from the computer's listed devices, and I can't reconnect the mouse at all (I've tried leaving it on and seeing if the connection reestablishes, getting it to "rediscover" the mouse from the mouse menu, and trying to get it to "find" the device through the device menu).

The only way I've found to fix this is to restart my whole computer, which is a pain. Is there any way to shut down/force quit Bluetooth (I tried finding the process in the Activity Monitor with no success), or at least force a disconnect with the device?