In the current stage, the values of Enum of "DocumentApp.ElementType" doesn't include the smart chip of the dropdown type. And also, for example, when the dropdown type is checked using "getType()", "UNSUPPORTED" is returned. Also, even when Docs API is used, it seems that the dropdown type cannot be retrieved.
From these situations, I think that in the current stage, unfortunately. the smart chip of the dropdown type cannot be managed using Google Apps Script. But, I believe that this will be resolved in the future update.