Thought I'd post an update to this topic. First, the addition of a "Move to..." choice in the later versions of the Right-click plugin (contextmenu) have reduced the pressure to implement this somewhat.
I was unable to get both a menu dropdown and the drag-n-drop working at the same time. So my workaround was to implement this as a skin choice, as shown here.
Steps to implement this are as follows.
1.) Edit the labels.inc file in whatever languages you use. The label to change is $labels['skin']. In my case, I changed it to:
$labels['skin'] = 'Move messages via';
2.) Copy the default skin folder to a new one for the skin with the menu option. For example, in Linux and using English, I did:
cp -Rp skins/default skins/Select\ in\ Toolbar
3). Rename the default skin folder to a name that makes sense to the normal behavior. In my case, I did:
mv skins/default skins/Drag\ and\ Drop\ (Default\)
4.) Edit skins/Select\ in\ Toolbar/templates/mail.html, search for the following line:
And add the line shown here just above it, like so:
Lastly, note that if any of your active plugins have skins folders in them, you may need to copy and rename to match the names of the skin selections in the master skins folder. The contextmenu plugin is an example, I had to copy the plugins/contextmenu/skins/default plugin and rename them to match the above to get the little menu icons to load.