Worked with a friend of mine and figured it out. For posterity: due_date is never actually blank, which is why the normal is_blank would not take. due_date defaults to 1. So in bug_api in the validate function:

    if( $p_update_extended ) {
        # due_date field cannot be empty
        if( ( $this->due_date == 1 ) ) {
            error_parameters( lang_get( 'due_date' ) );
            trigger_error( ERROR_EMPTY_FIELD, ERROR );
        }
    }

For good measure, updated bug_report_page with the required * as well:

<span class="required">*</span>

Please close out as needed.

Writing a custom validation function would also fix the issue without changing existing MantisBT code.