I think there's at least one degree of freedom there: You can raise (or lower) any class of vertex to change the dihedral angles. But for the "roundest", perhaps they need to be equal? (How is "roundest" defined? Minimum surface area for a fixed volume? Minimum ratio between min and max radius?) For what it's worth, I too chose equal dihedral angles for my polyhedron database: https://www.karzes.com/polyhedra/polyhedron.html?ph=V4.6.10 (If you have a mouse, you can click-and-drag to manually rotate.) Tom Bill Gosper writes:
The roundest polyhedron in PolyhedronData: In[86]:= MaximalBy[# | (# | N@# &@Min@PolyhedronData[#, "DihedralAngles"]) & /@ PolyhedronData[], #[[2, 2]] &] // tim
During evaluation of In[86]:= 0.537811 (* seconds *),1 (* winner *)
Out[86]= {"DisdyakisTriacontahedron" | (ArcCos[1/241 (-179 - 24 Sqrt[5])] | 2.87783661046122)}
{. . . | Sharpest dihedral}
In[87]:= Labeled[PolyhedronData@%[[1, 1]], %[[1, 1]]] Out[87]=DisdyakisTriacontahedron <http://gosper.org/D120.png> In[91]:= PolyhedronData@120
Out[91]= {"DisdyakisTriacontahedron", "IcosahedronSixCompound", {"IcosahedronStellation", 3}, {"IcosidodecahedronStellation", 1}}
In[92]:= Tally[PolyhedronData[%[[1]], "DihedralAngles"]]
Out[92]= {{ArcCos[1/241 (-179 - 24 Sqrt[5])], 180}}
claims that all 180 edges (60 short, 60 medium, 60 long) have this same angle. Is this obvious?
In[95]:= PolyhedronData[%91[[1]], "AlternateNames"]
Out[95]= {"28\[Hyphen]uniform dual polyhedron", "hexakis icosahedron"} —rwg