I rather like @LukeD's answer, but I started writing this last night, and I like that it starts from where @Accumulator left off:
I'm no expert with the bevel tool, so this might not qualify as the answer to your original question, but the question I saw at the end was how to produce the desired shape. That I think I have a good answer:
For reference, I've labeled the points in your last screenshot:
- Select and delete vertex A and vertex D.
- Select vertices L and R, and snap the 3D cursor to the selection, so it should jump to C.
- Select vertices L and B, duplicate them with [Shift]+[D], and rotate them 45 degrees around the 3D cursor.
- Add vertices L and B to the selection, duplicate them with [Shift]+[D], and rotate them 22.5 degrees around the 3D cursor.
Steps 3 & 4 gave you 6 new points. These are where we will move points E, F, G, J, K and M to using this procedure 6 times:
A. Select the new vertex location.
B. Add the original vertex to the selection.
C. Merge the vertices to the first point.
(Optional) move vertex H and vertex N slightly closer to C, to ensure all quads are properly convex.
This diagram should give an idea what the final quads would look like:
I've been away from blender for a few weeks, so feel free to add any hotkeys I may have forgotten. I hope you find this helpful.