Not really sure what you're trying to do... but if the idea is to loop over the keys of either the %dat_mapping or the %eng_mapping hash, depending on whether $hash_name is 'dat', then you could do

#!/usr/bin/perl
use strict;
use warnings;

my %dat_mapping = (foo => 1, bar => 2);
my %eng_mapping = (FOO => 3, BAR => 4);

my $hash_name = "dat";

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

            print "$file_name\n";
}
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See Conditional Operator