Regarding the expanding cob hat with surfaces facing down, my
experience is that areas of cob that are significantly farther from the heat source will hardly ever feel warmer than the general room. My bell has around 6" of cob over a brick inner skin, and at the corners (even though they are rounded to 6" radius) the surface does not begin to get warm until 6-8 hours of continuous burning.
The one way an overhanging hat could be beneficial is that rising heat from the sides of the barrel would hit the hat surfaces and warm them, reducing the amount of instant convective heating in the space and converting it to stored heat.