How to connect a bluetooth mouse without using the cursor (chicken and egg problem)

The trouble is with the devices, the Apple mouse and the keyboard in your case. They don't pair with more than one BlueTooth host at a time. You need a BlueTooth host that's "central" to both Macs if you want to use them.

You can find KVM switches that have BlueTooth host receivers in them. For example, the Zonet KVM3322W (dead link) worked over USB and provided BlueTooth receiver capabilities. I cannot attest to how well that would work. Presumably you'd pair the keyboard and mouse with the BlueTooth receiver in the KVM instead of in either Mac and then using the keyboard shortcuts, switch between control of the Macs who are connected to the KVM via USB. How the Macs would respond to the BlueTooth receiver in the KVM coming and going as you switched between them is unclear to me. Could be they handle it just fine.

The other option is to seek out BlueTooth accessories that allow themselves to be paired with more than one host. And then switch between those hosts from hardware keys on them. For example, this keyboard+trackpad combo from IOGear supports switching between 6 BlueTooth hosts. So you'd pair it with both Macs and the use the keyboard switches to decide which one you were talking to at any point in time. That, to me, seems like the more reliable approach. Of course: now you're stuck using IOGear's keyboard instead of the Apple peripherals.

Check the battery level in your keyboard or mouse. I've found this happens consistently when the peripheral battery is running low and the device seems to start spamming the Bluetooth connection with "help I'm running out of juice" requests. Swapping out your batteries for full ones will reliably fix the issue.