As I understand it, the TestComplete event for Unexpected/Overlapping window triggers if you attempt to interact via mouse click or keystroke with a component on the underlying window. So, whether or not the component is visible on screen shouldn't matter to TC. Do you have code or something in your tests to check for "VisibleOnScreen" before proceeding? If so, you may want to modify that to only check the "VisibleOnScreen" property when it is necessary to do so.
In any case, once the unexpected/overlapping window event gets triggered, you can create an event handler that will do the necessary button click as you described. See https://support.smartbear.com/viewarticle/84027/ for more information.
