@Mizar: I fixed the answer (better late than never, I suppose), although it does make the answer rather more complicated.

I'll come back and add the proofs of (2) and (3) when I have some time.

As I mentioned in my comment to the math.stackexchange question, an isometry of GH must fix the single point space and also the diameter of any space. Furthermore, using similar ideas, it can be shown that the number of points in a space is invariant under isometry in GH. From there, it seems likely that the only isometry is the trivial one.

Thanks! I'm still interested in how high the lower bound can be. So we've narrowed to between 0.032 and 0.22