The built-in kvm feature is very basic and I don't even setup the keyboard shortcuts because that requires software to be installed on the computers. Basically it just lets me switch between computers with 2 quick button presses on each monitor. I use an additional usb 3.0 switch from CableMatters so my peripherals can be switched between each display incase I want to have both computers up side by side.
If you only need 4K, there is this extremely badass TESmart KVM switch that I have used before and would highly recommend. It also has a cycle mode. It's $700 but the thing supports 3 4K monitors and 4 computers and works exactly as described. https://www.tesmart.com/products/4-port-triple-monitor-hdmi-...
The best option I've found is to hook both computers up to a USB-only KVM, and plug them in via separate video cables to the monitor. Then set up the computers to tell the monitor to change inputs (via DDC commands, which most but not all monitors support) when you change devices on the KVM
I don't think any KVM supports Bluetooth. You either need to use software like Barrier or get devices that can switch between multiple. The other option is wireless peripherals that use a USB dongle can work fine with KVMs like the ones Logitech sells.
If you only need 4K, there is this extremely badass TESmart KVM switch that I have used before and would highly recommend. It also has a cycle mode. It's $700 but the thing supports 3 4K monitors and 4 computers and works exactly as described. https://www.tesmart.com/products/4-port-triple-monitor-hdmi-...