By design RadAutoCompleteBox does not have a DropDown button. Have you made some modifications to the control?
We have an SDK example in our Github repository that demonstrates how to populate the drop down portion of the AutoCompleteBox with items uwing a Button:
We have noticed that the example does not behave as expected, so we have modified it in order to demonstrate you how to open and close the DropDown using the Button - please, check the attached file.
Hope this helps.
Telerik by Progress