From what I gathered from the docs and a lot of testing, it would look like

use IPC::Run qw( run );

open(my $fh, '>>', '/Volumes/Expansion Drive/stuffTGZ/list_xxx_dirs.tx
+t') or die $!;

print($fh "-------------------------------------------------\n");
print($fh "- $File::Find::name -----------------------------\n");

run
   [ "/bin/ls" => ( "-l", $File::Find::name ) ],
   '<', \"", '>', $fh;
[download]

Notes:

• Lexical file handles work, but I didn't see it in the documentation.
• «'<', \""» appears to open /dev/null (desirable) as opposed to closing STDIN as documented (undesirable).
• The child appears to inherit the parent's STDERR if 2> isn't specified. That's probably documented, but I didn't see it.