Usually the Open, Save & Cancel pop-up comes from the browser, what is the status code you get from LWP? In any case the following snippet should get the file for you:

use strict;
use LWP::UserAgent;

my $ua = LWP::UserAgent->new;
#my $req = HTTP::Request->new(GET => 'http://search.cpan.org/CPAN/auth
+ors/id/G/GA/GAAS/libwww-perl-5.836.tar.gz');
my $req = HTTP::Request->new(GET => 'http://usename:password@host/path
+/to/file.zip');
my $res = $ua->request($req);

if ($res->is_success) {
    # here put what you want to do with the file
    # print $res->content;
    # or
    # open( my $file, '> file.zip');
    # print $file $res->content;
    # close($file);
} else {
    print STDERR $res->status_line;
}
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