So does this mean that the content-length header value is no longer used?
Here is some untested sample code based on your response and reading the lwp::useragent docs...

use strict;
use warnings;
use LWP;
use IO::Handle;
use Fatal qw( open close );

my $url = 'https://........';
my filename = 'file2send.txt';
my $LINES_PER_CHUNK = 100;
open my $fh, '<', $filename;

sub sendthis {

    my @buf;
    my $linesread = 0;

    if ($fh->isopened) {
        while (my $line = <$fh>) {
            $linesread++;
            push(@buf, $line);
            if ($linesread == $LINES_PER_CHUNK) {
                return join('', @buf);
            }
        }
        close $fh;
        return join('', @buf);
    }

    return '';    
}

eval {
    my $headers = HTTP::Headers->new();
    $headers->header('Content-Type' => 'text/plain; charset=utf-8');
    $headers->header('Content-Length' => -s $file2send);
    my $request = HTTP::Request->new('POST', $url, $headers);
    $request->protocol('HTTP/1.1');
    my $browser = LWP::UserAgent->new();
    my $response = $browser->request($request, \&sendthis);
    my $gotthis = $response->content();
    print $gotthis;
};
if ($@) {
    # handle error !
}
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Edit: made changes that rhesa suggested regarding request method callback and content length