Now to something completely different.

Did you think about the inverted problem to construct sets with a maximal hamming distance?

Those sets are much smaller, i.e. any n-set has either a span of n or is reasonably small.

For instance the biggest 3-set of span 2 is

AAA
ABB
BAB
AAB
[download]

And it doesn't matter how variable your selection is, another C or D at one position doesn't change the fact that any set with more than 4 selections must have at least 2 selections with distance 3. ¹(see update)!

Look for "hamming codes" for formulas to estimate the maximum bound.

So no need to test all pairs if the number gets bigger.

This should shorten calculations considerably...

HTH! :)

¹) hmm sorry there is one edge case I forgot...

AAA
ABB
ACC
ADD
AEE
...
[download]

But you can easily see that the position with the smallest variety already limits the maximal set. And it's very easy to check.