From you description, the only "sync" criteria betwen the files seems to be the sequence.

Based on that, this should work (untested):

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

open my $dnaseq, '<', "FILE2.TXT" or die "Can't open FILE2(Sequences):
+ $!\n";
open my $namesfile, '<', "FILE1.TXT" or die "Can't open NAMES: FILE1: 
+$!\n";
while (my $name = <$namesfile>) {
   chomp $name;
   <$dnaseq>; # Throw away "BARmmm;size=xxx" record
   chomp(my $seq = <$dnaseq>) ;
   # Should check if we actually have a $seq here....
   print "$name\n$seq\n";
}
close $dnaseq;
close $namesfile;
# Should check if there are left over records in $dnaseq....
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