See, the problem is that most people want to be able to have a BT mouse/computer pair next to another BT mouse/computer pair without the mice or the computers getting confused. As a result, BT forces one-to-one devices like mice to be uniquely paired to one and only one system. When you move it to another system, it loses the pairing with the first one.
So, AFAIK, you'll need to keep resetting it. However, as you probably know, the easiest way to do so is to keep the BT status in the menu bar (System Preferences...->Bluetooth->Settings->"Show Bluetooth status in the menu bar").
Once that's set up, then, for each time you need to reset the mouse, in the menu bar, click on the BT icon and go to "Set Up Bluetooth Device...". It should only take a few seconds to set up your mouse - and locking it to one computer should stop it from talking to the other.
At least, in theory this should work... I have BT, but don't have a BT mouse (use it with cell phone and keyboard only).