Encoding the strings manually and then outputting them with syswrite() works too:

use strict;
use warnings;
use 5.010;

use Encode qw{encode};

my $string = "Line1\015\012Line2\015\012\015\012Line4\015\012";    
my $utf16_string = encode('UTF16-LE', $string);

open my $OUTPUT_FH, '>', 'data1.txt' 
    or die "Unable to open data1.txt: $!";

syswrite $OUTPUT_FH, $utf16_string ; 



--hex output:--
4C 00 69 00 6E 00 65 00 31 00 0D 00 0A 00
4C 00 69 00 6E 00 65 00 32 00 0D 00 0A 00 0D 00 0A 00
4C 00 69 00 6E 00 65 00 34 00 0D 00 0A 00
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