A "stable shuffle" is just an arbitrary arrangement that doesn't change. So, why not just produce one and then embed it in the script(s) that requires it. Using your example above:

C:\test>perl -MList::Util=shuffle -wle"print join ' ', shuffle 'A'..'Z
+'"
Q I X H L R B C F E U M A S W J P K T O V N Y D G Z
[download]

And then just c&p that into your code:

...
my @arr = qw[ Q I X H L R B C F E U M A S W J P K T O V N Y D G Z ];
...
[download]

There you have a "stable shuffle" that stays stable until you change it. Without messing around with storing it in a DB or file.

(I also find your reasoning for not using srand inexplicable, but that's been said, and this is simpler:)