I've made some headway on the problem, and now seem to be getting mail through to Earthlink. While digging around for information, I found an old post (via Google) that suggested that trying to pass a full email address to MIME::Lite could cause envelope problems. Sure enough, as I examined messages, I found that messages sent via

  my $msg = new MIME::Lite(
      From => "Registration System <register\@example.com>",
      ...
[download]

resulted in an email envelope and header that looked like

  From Registration-System-<register@example.com Fri Sep 26 18:36:37 2
+003
  Return-Path: <"Registration System <register"@example.com>
  ...
  From: Registration System <register@example.com>
[download]

which definitely looks bogus. Changing to

  my $msg = new MIME::Lite(
      From => "register\@example.com",
      ...
[download]

produces a sensible envelope and header, and Earthlink seems to like it.

Looks like it's time for an archeological dig into MIME::Lite.