Likely the doubling of the disc is the best. Who can do better?
At least, if the diameter of the doubling is 1 then the average distance is bigger than $\tfrac12$. This shows that the optimal surface is not the round spher (i.e., the symmetry is not maximal). In this case, the chances that you will get an answer are nearly 0.
For example, the is a closely related problem of Alexandrov to maximize area among all the positively curved surfaces with diameter 1. The same picture --- round sphere is not the answer, the doubling of the disc is likely the best and the problem i open for more than 50 years.