I'd change this arround. Give the "non browse" field a non transparent fill color and put it on top of the drop down field to hide it. Clicking on this field will cause the drop down behind it to pop to the front and deploy the valuelist. When you exit the field, it will disappear back behind the other field.
PS. Since you don't want to show the arrow, remove that option from the field's format in the inspector. That will help the drop down deploy more easily.