Switching Logitech mouse and keyboard between two computers (Logitech Flow not working since work computer is on a VPN)

This question is open for many opinions and different answers, so here goes my 2 cents.

Since you cannot have access to your local network from your work computer you need to have a hardware based solution:

Buy a new Logi Keyboard/Mouse combo with switch buttons:

IMO this is your best option. Personally I have a MK850 Keyboard and Mouse Combo (K850 keyboard + M720 mouse) and both are compatible with more than one Unify USB receptor (also Bluetooth without a new dongle). They have special key/buttons to switch from one receptor to another, but you'll need to press it on the keyboard AND on the mouse - if don't, keyboard could be attached to one receiver and mouse to another. But it is a pretty simple design, you'll get used to it.

Buy an USB hardware KVM

This reminds me the old days on Tower servers without special server KVMs. You can plug your single USB keyboard and USB mouse of any brand into the available "client" USB ports and connect the "server" USB ports on each computer. Then you simply press one button on the KVM to switch from one to another. All USB devices connected into "client" ports will be attached to only one computer at time.

I have this one https://shorturl.at/ewKQ5 (link to AliExpress) before buy the MK850 combo and it worked flawlessly.