Mac – How to pair the Magic Trackpad with multiple Macs

I have a Magic Trackpad that I'm using with my iMac.
Everything works great, but I would also like to use the trackpad with my MacBook.

For some reason, it seems that I can only pair the trackpad with one computer at a time.
With the iMac turned off, I turned on the MacBook and went to set up a new Bluetooth device. The trackpad was not found.
I know that nothing is wrong with my connection procedure (I pressed the power button, it was within range, etc), and that nothing is wrong with the trackpad itself (it works fine with the iMac).

I finally got the trackpad to pair with the MacBook by unpairing it with the iMac, turning the iMac off, and power cycling the trackpad.

Then, when I went to re-pair with the iMac, the device was not found (the MacBook was completely off; it was not connected to anything). Only by unpairing with the MacBook and power cycling was I able to re-pair with the iMac.

So, I think the trackpad is trying to tell me that it can only be paired with one computer at a time.

I would like to pair the trackpad with both my iMac and my MacBook not for simultaneous use, but for using with one when the other is turned off or far away.
I have other Bluetooth devices that let me do this just fine, and I can't think of a reason why it would be different for the trackpad.

How can I pair my Magic Trackpad with both my iMac and MacBook for non-simultaneous use?

Ideally without any third-party software.