My thanks to all who provided help and assistance with this. I now have a working test solution which I present below as this may be useful for others.

I used Path::Tiny, as originally shown by ++tybalt89, to generate the list of files for archiving. I needed to tweak the suggested code in a few places, as follows:

• Path::Tiny's visit() method "Executes a callback for each child of a directory.". This meant that the parent directory wasn't added to the array for add_files(): easily fixed by preloading that array with the parent directory.
• For my purposes, the contents of the directory structure was well known, so no filtering — based on file types, permissions or other characteristics — was required. In other situations, some filtering may be required: see ++tybalt89's and ++Tux' examples^[1,2] for possible ways to achieve this.
• I also needed to change directories to get the exact archive I wanted. This may not be necessary in other circumstances. Note the use of the autodie pragma in a limited, lexical scope to handle problems with chdir and provide useful feedback if they occur.
• Not an actual tweak required by this code, but use 5.016; was added because it mirrors the version I'm coding to for $work$ (and increases confidence that this won't have issues in my production environment). If you're using this code as a template, and not adding anything fancy, replacing that with a simple use strict; would probably be fine.

Here's the test script. Note that the directory structure under $src_dir is exactly the same as that presented in the OP.

#!/usr/bin/env perl

use 5.016;
use warnings;

use Archive::Tar;
use Cwd;
use Path::Tiny;

my $src_dir = '/Users/ken/tmp/test_arch/src';
my $zip_dir = '/Users/ken/tmp/test_arch/zip';

my $top_dir = 'fred';
my $zip_name = 'fred.tar.gz';
my $zip_path = path($zip_dir, $zip_name);

{
    use autodie;
    my $cur_dir = getcwd;
    chdir $src_dir or die;
    my @tar_files = ($top_dir);
    path($top_dir)->visit(
        sub { push @tar_files, "$_" }, { recurse => 1 }
    );
    my $tar = Archive::Tar::->new();
    $tar->add_files(@tar_files);
    $tar->write($zip_path, COMPRESS_GZIP);
    chdir $cur_dir or die;
}
[download]

A sample run, as well as various checks, are in the spoiler: