Looking at LWP::Simple sources, it's already done:

sub getstore ($$)
{
    my($url, $file) = @_;
    my $request = HTTP::Request->new(GET => $url);
    my $response = $ua->request($request, $file);

    $response->code;
}
[download]

And $file ends up in LWP::Protocol collect() method:

        elsif (!ref($arg) && length($arg)) {
            open(my $fh, ">", $arg) or die "Can't write to '$arg': $!"
+;
        binmode($fh);
            push(@{$response->{handlers}{response_data}}, {
                callback => sub {
                    print $fh $_[3] or die "Can't write to '$arg': $!"
+;
                    1;
                },
            });
[download]

Files are not loaded into memory. You may want to tweak :read_size_hint argument of get() method of LWP::UserAgent if it still overflows.