You can do it with a plugin. I threw this together rather quickly. I don't do anything special for multiple cursors (though it should take the content of multiple cursors as well as paste to all the proper locations).
import sublime
import sublime_plugin
class PasteAndCopyCommand(sublime_plugin.TextCommand):
def run(self, edit):
view = self.view
current_content = []
cursors = view.sel()
for cursor in cursors:
current_content.append(view.substr(cursor))
for cursor in cursors:
view.replace(edit, cursor, sublime.get_clipboard())
sublime.set_clipboard("\n".join(current_content))
After you save the plugin, use paste_and_copy
as the the command for your key binding or command palette entry.