Hello pklgn,
Welcome to Microsoft Community.
The support link for the Microsoft Designer Compact Keyboard does state that the keyboard can be paired with three devices at the same time:
Use Microsoft Designer Compact Keyboard - Microsoft Support
Under normal circumstances, you can really switch between three devices with Fn+F1.
However, if this problem occurs at the moment, I recommend removing the Microsoft Designer Compact Keyboard from all three computers, and then pairing them separately by following the instructions above.
You can also test if there is an exception when pairing only two devices, such as 1-2, 1-3 and 2-3, to confirm that the problem is related to a specific computer.
Best Regards,
Mitchell | Microsoft Community Support Specialist