I did something similar last year with a monitor without built-in KVM but with good DCC support (Ultrasharp U3417W) and Synergy [0].

I use Synergy as part of my desk setup already, but needed a way to view the UI of a normally headless machine. The solution I built was a small shell script that terminated the active Synergy session and started a new one with a different config file (so keyboard/mouse input would map to the normally-headless machine), and fired off a DCC command to the monitor to change its input. The same script ran with a different argument would switch back to the normal display/control configuration. This solution worked pretty well until I was able to retire the headless machine early this year.

[0] https://symless.com/synergy

does synergy works better now? 3 years ago, every week I would get into a situations where one machine was not connecting to the other, and I had to randomly restart synergy so maybe it connects. fun to do that 5 min before the meetings.