(I'm replying to my own post because I believe I've found the best general solution to my problem.)

Once I'd figured out that I could use DateTime::format_cldr() to render the timestamps in the required format, it seemed incongruous to be parsing them using DateTime::Format::Strptime. The new carpet made me realize I needed to replace the drapes. I was doing this…

        pattern => '%Y%m%d %H:%M:%S',

…and this…

    my $pattern = 'M/d/y h:mm:ss a';

…in two successive statements, which is inelegant. So I decided to use CLDR to parse the timestamps as well as to format them. Doing this neatens and tightens the code.

#!perl

use v5.14;
use strict;
use warnings;

use DateTime;
use DateTime::Format::CLDR;

my $parser = DateTime::Format::CLDR->new(
    pattern  => 'yyyyMMdd HH:mm:ss',
    on_error => 'croak',
);

my $format = 'M/d/y h:mm:ss a';

while (my $timestamp = <DATA>) {
    chomp $timestamp;

    $timestamp = $parser->parse_datetime($timestamp)
                        ->format_cldr($format);

    say $timestamp;
}

exit 0;

__DATA__
20040805 18:31:00
20050106 10:54:27
20050302 01:23:35
20100808 20:00:16
20110501 18:09:44
20110909 20:02:42
20130408 18:09:03
[download]

This prints…

8/5/2004 6:31:00 PM
1/6/2005 10:54:27 AM
3/2/2005 1:23:35 AM
8/8/2010 8:00:16 PM
5/1/2011 6:09:44 PM
9/9/2011 8:02:42 PM
4/8/2013 6:09:03 PM

Jim