Bluetooth mouse vs 2x computers

I have a Bluetooth mouse (Razer Orochi), and I regularly switch between two computers which I'd like to use it with. Is there a way to make it so that I don't have to re-pair it every single time I change from one to the other?

There are two ways to do this.

The "old-fashioned" way of using a KVM is one. Essentially it's a box that you plug your devices into. Flip the switch on it to switch computers. There are other features provided (more than 2 computers, more than 2 in/outputs, buttons, yadda yadda yadda not relevant) but that's basically it.

You can also use Synergy, where basically you just move your mouse between screens to change both the "focus" of the keyboard and monitor. It works by pairing the device with a "host" computer then having it essentially pass off the input commands to the other computer when the mouse changes monitors.

ok a stupid solution I use is to disable any onboard bluetooth and use a dongle. Plug it into computer 1 - mouse and keyboard work on computer one - take it out and plug it into computer 2 - mouse and keyboard work on computer two.

Use Synergy

Synergy is Software that lets you easily share your mouse and keyboard between multiple computers, where each computer has it's own display. No special hardware is required, all you need is a local area network. Synergy is supported on Windows, Mac OS X and Linux. Redirecting the mouse and keyboard is as simple as moving the mouse off the edge of your screen.

I have not had direct experience with this product ...

Logitech Ultrathin Touch Mouse T630

but I do own it's sister product for keyboards ...

Logitech Bluetooth Illuminated Keyboard K810

I like this solution because is works with PC, Mac, iOS, and Android. I use an iPad at work that is not connected to my LAN. It uses cellular for data. I also have my MacBook Pro which is also connected to a different network. Then my development PC which is on the company network. I can easily switch back and forth between the machines as needed using one keyboard. It can pair with up to three (3) devices while the mouse pairs with two (2) devices at a time.