I don't see a likely property on either the StatusStrip or the ToolStripDropDownButton - other than the Alignment property of the ToolStripDropDownButton, which doesn't help.
Will I have to add "shims"/spacers or some such to elbow the ToolStripDropDownButton away from the left edge?
I don't know if this is the best way to do it, but I added a toolStripStatusLabel, and set its Spring property to True and its Text property to [blank].
It works, so I guess it's good...