I've encountered this issue as well. It could probably be solved if the dropdown button offered some kind of "selectedItem" field so that code could access whichever option the user ended up clicking.
When Dropdown Button or Split Button has its actions configured in "Mapped" mode the item that was clicked as well as its index (as i) are available in Event Handlers.
Here's an example of how an array of strings could be mapped to URLs:
However, our documentation is unclear, and it looks like this is not working for the "Run script" Event Handler. I'll file a bug now and will report back when this is resolved.