It may be easier to have just the domain name passed in to the script and then the script can handle ALL of the prefixes itself.

Yeah, that sounds sensible.
Here is an example of how you might implement that approach:

#!/usr/bin/perl
use strict;
use warnings;
use LWP::Simple;

DOMAIN:
while (my $domain = <DATA>) {
    chomp($domain); 
    for my $protocol (qw/http https/) {
        next DOMAIN if test_url("$protocol://$domain");
        for my $sub (qw/www web/) {
            next DOMAIN if test_url("$protocol://$sub.$domain");
        }   
    }   
    print "Couldn't get anything from $domain\n";
}   

sub test_url {
    my $url = shift;
    print "Trying $url ...";
    my $ua = LWP::UserAgent->new(
        timeout     => 5,
        agent       => 'Mozilla/5.0',
        ssl_opts    => { verify_hostname => 0 },
    );  
    my $response = $ua->get($url);
    if ($response->is_success) {
        print "OK\n";
        return 1;
    }   
    else {
        print "FAILED because " . $response->status_line . "\n";
        return undef;
    }   
}   

__DATA__
google.com
apple.com
fred.com
dschjksdbckjqh.com
[download]

Trying http://google.com ...OK
Trying http://apple.com ...OK
Trying http://fred.com ...OK
Trying http://dschjksdbckjqh.com ...FAILED because 500 Can't connect t
+o dschjksdbckjqh.com:80 (Bad hostname 'dschjksdbckjqh.com')
Trying http://www.dschjksdbckjqh.com ...FAILED because 500 Can't conne
+ct to www.dschjksdbckjqh.com:80 (Bad hostname 'www.dschjksdbckjqh.com
+')
Trying http://web.dschjksdbckjqh.com ...FAILED because 500 Can't conne
+ct to web.dschjksdbckjqh.com:80 (Bad hostname 'web.dschjksdbckjqh.com
+')
Trying https://dschjksdbckjqh.com ...FAILED because 500 Can't connect 
+to dschjksdbckjqh.com:443 (getaddrinfo: nodename nor servname provide
+d, or not known)
Trying https://www.dschjksdbckjqh.com ...FAILED because 500 Can't conn
+ect to www.dschjksdbckjqh.com:443 (getaddrinfo: nodename nor servname
+ provided, or not known)
Trying https://web.dschjksdbckjqh.com ...FAILED because 500 Can't conn
+ect to web.dschjksdbckjqh.com:443 (getaddrinfo: nodename nor servname
+ provided, or not known)
Couldn't get anything from dschjksdbckjqh.com
[download]

HTH,
Darren