You said that user1@FQDN worked from the FTP client but you didn't say that you've tried it from your Net::FTP script. Also, double check that you have the right password in your script.
my $ftp_site = '126.96.36.199 5000';
my $ftp_dir = '/Directory2';
my $ftp_get_file = 'that.file';
my $ftp_user = 'user1@FQDN'; # if 'DOMAIN/user1' and 'DOMAIN\\user
+1' don't work, perhaps this will
my $ftp_password = 'password321'; # double check your password, maybe
+it's the problem!
my $ftp = Net::FTP->new($ftp_site)
or die "Could not connect to $ftp_site: $@";
or die "Could not login to $ftp_site with user $ftp_user: ", $ftp->
or die "Could not change remote working directory to $ftp_dir on $f
+tp_site: ", $ftp->message;
or die "Could not get $ftp_get_file the file is not present in $ftp
+_dir: ", $ftp->message;