Daniel,
It is possible to move an overlay layer above graphics layers, however it is accomplished outside of the jsapi, and requires some advanced knowledge of the dom and dojo to accomplish it. Unfortunately doing so will cause several problems.
Here's an image of the map's dom node from your fiddle. As you can see graphic/feature layers exist in a dom node at the same level as overlay (dynamic, tiled, web tiled, wms, etc) layers. When you reorder layers, if it's an overlay layer it's reordered with the other overlay layers. If it's a graphic/feature layer it's reordered within the graphics dom node.
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You can use dojo (or pure js) to move an overlay above (it's reversed in the dom - top most dom node is the lowest layer) the dom node containing all the graphic/feature layers. HOWEVER, doing so will "cover up" the graphics layers and you WILL NOT be able to interact with them, such as clicking, which in necessary to show the popup for example. And if you try to reorder any layer with the api's reorderLayer() after having done so, the map will crash.
Hopefully, that explains what's going on and why you shouldn't do it. As one solution to your problem, take Ken's example and simply add a click event to the map to identify, highlight (with map.graphics) and show popup using your layer.