If a KIS inventory part is copied via "alt+click" action in the editor, the inventory content doesn't get copied.

Alas, this is not feasible. KSP editor copies a Unity object with only few part's properties. It doesn't let capturing the original part's state to replicate it in the target. There can be workarounds made, but none of them would be reliable. The stock cargo copy logic suffers from the same problem.