The final key here was that the second argument to the post method needed to be an array ref (i.e., the hash arguments needed to be enclosed in []s). The code below works, returning the desired output page in the $mech object.

#!/usr/bin/perl
use warnings;
use strict;
use WWW::Mechanize;

my $hypermutUrl =
  "http://http://hiv-dev.lanl.gov/cgi-bin/HYPERMUT/hypermut.cgi";
my $mech = WWW::Mechanize->new( autocheck => 1 );

my $alignment_as_string = <<'END_SEQS';
>HIV1-test.CONS
ATGGGATGTCTTGGGAATCAGCTGCTTATCGCGCTCTTGCTAGTAAGTGCTTTAGAGATTTATTGTGTTC
>HIV1-test.1
ATGGAATGTCTTGGAAATCAGCTGCTTATCGCGCTCTTGCTAGTAAGTGCTTTAAAGATTTATTGTGTTC
>HIV1-test.2
ATGGAATGTCTTGGGAATCAGCTGCTTATCGCGCTCTTGCTAGTAAGTGCTTTAAAGATTTATTGTGTTC
END_SEQS

$mech->post(
    $hypermutUrl,
    [
     'FORMAT'            => 'FASTA',
     'ALIGNMENT'         => $alignment_as_string,
     'upfile1'           => '',
     'INN'               => '',
     'OUT'               => '',
     'MUTUPSTREAM'       => '',
     'MUTFROM'           => 'G',
     'MUTTO'             => 'A',
     'MUTDOWNSTREAM'     => 'RD',
     'ENFORCE'           => 'DESCENDANT',
     'CONTROLUPSTREAM'   => '',
     'CONTROLFROM'       => 'G',
     'CONTROLTO'         => 'A',
     'CONTROLDOWNSTREAM' => 'YN|RC',
     'Analysis'          => 'All',
     'submit'            => 'Run',
    ]
);

my $page = $mech->content;
open my $FH, ">/tmp/hypermut.html";
print {$FH} $page; close $FH;
warn "saved webpage data to /tmp/hypermut.html\n";
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Thanks to any and all who thought about it, and most particularly to my colleague who found my error.