I've done exactly the same thing with a client company in the past. In that case it involved transferring sensitive banking data to laptops which were to be used by the client's representatives in the field.
My preferred method, given the need for avoiding removable media,was the one you describe in your final paragraph; we instituted a policy of copying to a dedicated desktop and using a single authorised member of staff to disconnect the desktop from the network after the copy process, copying the data to the laptops, disconnecting them, deleting the desktop copy and reconnecting the desktop.
It worked smoothly, and was very secure, given that one person handled the process. As far as I know the client still follows the same procedure.