The best approach according to the docs is to override the get_basic_credentials method of LWP::UserAgent (WWW::Mechanize's parent class) so that it returns a username and a password. So I'd try something like this (untested):

package Nonce;
use base 'WWW::Mechanize';
sub get_basic_credentials { return qw( user password ) }

package main;
my $M = Nonce->new();
$M->get('http://anywhere.com/mytest?rm=test', @args);

# etc.
[download]