• key should be keys
• You never call the anon sub. ->() can be used to call it.
• I don't see why you're using a sub at all.
• Your anon sub returns a reference to a hash, not a hash, so you need to dereference it.

for my $file_name (
   keys %{
      $hash_name eq 'dat'
         ? \%dat_mapping
         : \%eng_mapping
   }
) {
    ...
}
[download]

It would be more readable to break it up:

my $hash = $hash_name eq 'dat'
   ? \%dat_mapping
   : \%eng_mapping;
for my $file_name (keys %$hash) {
    ...
}
[download]

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