I currently have this result:
Here are my Appearance window settings:
I created it by putting the Round Corners on the clipping mask.
As you can see, in addition to the Round Corners there is also supposed to be a navy blue Stroke, but it's not showing up.
I have tried dragging the Stroke up and down, changing its order in the Appearance window. Still nothing.
The Opacity is set to Default, so it's not invisible due to it being transparent.
Here is a simplified example of the kind of result I want:
But I got this result by creating a duplicate rectangle, not by adjusting the Appearance settings of the original clipping mask.
I would like to create this as an Appearance that I can then apply to many other clipped images.