Nick,
We were looking at adding this feature but ran into the same problem you are describing, e.g. when to include and when to not include headers.
Honestly we were waiting for people to start using the new UX for the differing copy and paste functionality and waiting to see what people said.
The current design (pgAdmin 4 1.5 +) the existing behavior is a hybrid of excel like functionality and a non-excel table like behavior. One option would be to go full Excel and treat the headers like a row.
I think my concern with adding a new keyboard shortcut for copy or copy with headers would be a discoverability issue.
-- Rob