Quote:
...Do you have a simple instruction just to move the selected beret onto the dumb politician's head? ...
Just the same as before. For this type of beginner question, it really is best if you give full screenshots of the Gimp interface. The tool options and layers dialogues often give clues as to what is happening.
example for moving a layer

1. make the layer active
2. Use the move tool
3. To prevent moving the base layer, enable move active layer
Now click-and-drag on the canvas.
Quote:
...When I try with the eraser nothing happens. I rub the eraser around the beret and there is no visible change..
I suggested using the eraser tool as a simple method, for a simple shape, for a beginner.
However since you made a selection the eraser tool (any paint tool) does not work outside the selection.
With the selection active, as your screenshot, invert the selection then cut the selection, then turn the selection off. Then move around as earlier.
As an example, to practice moving layers, something like your image, an xcf file attached.