I discovered this problem can be solved by encoding the email message in base64. CakePHP's email component does not natively do this, so I wrote a class Email64 to extend the email component. I rewrote all functions that contained
'Content-Transfer-Encoding: 7bit';
to
'Content-Transfer-Encoding: base64';
And then in the _mail() function, I replaced calls to php's mail function -- something like this --
return @mail($to, $this->_encode($this->subject), $message, $header, $this->additionalParams);
to this --
return @mail($to, $this->_encode($this->subject), rtrim(chunk_split(base64_encode($message))), $header, $this->additionalParams);