I apologize. You are right about the dropdowns. You cannot specify drop down fields for use in Calling Cards. This should actually be moved to the Feature Request Forum.
Then custom fields lets you put the fiels in password format, email format, # format, madatory and enabled. I cant see anything that says drop down menu. It is weird that it is available under the technician console but seems nowhere under the calling card.
I think what you are looking for is in the Admin Console under Calling Cards and the Custom Fields.