There is a huge difference in the app handling of a paste action and a drag and drop action.
So if FCP is expecting a Drag/drop, and does not accept a paste for the same thing, then you are going to have to do an actual drag/drop.
I don't think that is possible.
You can definitely use KM to conduct a drag/drop operation. You just need to determine how you want to specify the start/stop coordinates. Here is a macro that let you click at the start and stop points, and then does the drag/drop for you. You could also use found image, and relative position from a window corner.
MACRO: Drag/Drop Mouse To/From Points Selected by Mouse Position
Good luck, and let us know how it goes.