You might be interested to know, that, when $releasing$loosening the questcondition of being a regular polytope, then not only sporadically some oddballs do fit your property of having circumradius = edge length, but rather there is a series of Wythoffian polytopes, which throughout any dimension would have this property!
In fact that would be the expanded simplex with Dynkin diagram $x3o3o...o3x$ (where I've used typewriter friendly replacements, $x$ meaning a ringed node, $o$ represents an unringed node).
- $x3x$ is the hexagon
- $x3o3x$ is the cuboctahedron
- $x3o3o3x$ is the small prismated decachoron (aka: runcinated pentachoron)
- $x3o3o3o3x$ is the small cellated dodecateron (aka: stericated hexateron)
- etc.
Indeed all these have the common property that its equatorial section happens to be nothing but the former item of this list. In fact, in this very orientation they all can be described layerwise as a simplex atop the expanded simplex atop the dual simplex:
$$x3o3o3o...o3x = hull( x3o3o...o3o\ ||\ x3o3o...o3x\ ||\ o3o3o...o3x )$$$$x3o3o3o...o3x = \text{hull}( x3o3o...o3o\ ||\ x3o3o...o3x\ ||\ o3o3o...o3x )$$
--- rk
