Ah! I think I've got it (with dchetlin's help)...
The trick is to triple escape any $ characters

perl -pi.orig -e "s#<% \\\$flat->purge\('dynamix','outburst'\) %>#\\\$
+CONFIG{DYNAMIX_OUTBURST}#" www/index.html
[download]

This MAY be shell-dependant.
Anyone know any more about this?

Thank you! Note to self: "Boy, I love it when that dim bulb lights up. Must be Friday."