BIS disabled the save to clipboard on MP – so no. You can always save to profile.
Save to profile - well, it saves to profile - client.
So if you are playing on a server save to profile, then open a local host session then load if from profile and then save it to SQM.
I tried making a Base on my dedicated server then saved it with the MCC profile option. Next i started a new editor (local) map and tried loading the MCC profile. Nothing was added to the new map, none of the objects or players.