I contacted WPEngine Support and got my unique situation sorted out. Here’s what Spencer said:
I’d mention that the debug logging should be made a public feature, something that can be toggled on/off within Local instead of having to add defines to wp-config.php manually but also include that adding the defines I gave you will remove the warnings that aren’t critical errors that prevent the site from working properly.
The fix to get around those messages were to add this in your wp-config.php file, add it right under the table_prefix
ini_set('display_errors','Off');
ini_set('error_reporting', E_ALL );
define('WP_DEBUG', false);
define('WP_DEBUG_DISPLAY', false);
So hopefully this helps someone in the future who is seeing strange errors keeping them from getting their work done. All is well that ends well.
Thanks @ben.turner for getting me part of the way there.