I do have a backup, but the current situation is the best I could do to update it, based on the generic Joomla! documentation, and if I were to repeat the update process, I'd do just the same steps I've already done.
However if you have specific documentation for the corporate_response template about how to correctly upgrade it from Joomla 3.2.7 to 3.9.3, I'm more than willing to read and follow it.
BTW, I could not reproduce the problem you encountered, but I've now updated a number of links that were pointing to wrong URLs coming from the old permalink scheme that included index.php.