Sorry, but should have commented on this much sooner.

After my last comment I eventually replaced the ATEN CS1924 KVM with a new
generic 4-port KVM that is 4K@60Hz-capable. This one does not exhibit the
issue, so I could consider my own issue as having been worked around. The new
KVM costs about 1/4 of the ATEN KVM, though.

I'm not sure about the OP's situation or whether he had worked around or not,
but this kinda proved that the issue is indeed configuration-specific.