Note: The power plan does NOT put the secondary computer to sleep nor does it turn off the hard drives.
And if you have a performance power setting, where nothing is turned off, does that change the outcome?
While moving the mouse on the secondary computer does fix the issue, this is very problematic for secondary computers separated by many miles (e.g. vacation homes connected using VPN (so on the same subnet).
It just reads like something else is getting powered down.
When KVM'd to the PC, and with it having focus, see if you can jar it with a Ctrl + Alt + Backspace (that is the same as a Ctrl + Alt + Del on a local PC). At the very least, you should see the typical menu for that act.
Something else to try... While your PC needs to support and be set up to respond to Wake-on-Lan (WAL), try enabling:

Sean Drohan
Stardock Product Lifecycle Manager