Have you tried local $/ ? The regex seems to function fine if you flatten the file. This works for me:

use strict;
my($FILE,$SITE);
print "Protein Sequencer\n\nInput Dataset file name: "; 
$FILE = <STDIN>;
#chomp($FILE);

local $/; #Null the $/ to search through a flat file
print "\nLoading Dataset - Be patient.\n";
open DATA, $FILE or die $!;# Open File
$_ = <DATA>;# Load File to Ram
close DATA or die $!;# Close File

print "\"N\" Sites\n---------\n";
while ($_ =~ m/\w{1,12}\s+(\d{1,5})\s+[a-zA-Z]{4}/gs) {
 $SITE = $1;
 print "$SITE\n";
}
[download]