Is there a way to retain core search logging instead of having it get deleted with the watchdog logs?

We're using the core search module on a D9 site and use the "Top Search Phrases" report for search tracking but the results get stored in the watchdog table so they get cleared out when the logs do.

I see there are some D7 modules to improve the core search logs but nothing that I could find for D8.

The client needs to retain the search word logging so how could we do this on our D9 site?

I don't know if there exists a module (off-topic anyway on this site), but you could try to implement your own cron hook. First disable the built-in clean up process by setting "Database log messages to keep" to "All" in the UI.

Then implement your own hook_cron(), similar to the core hook

/core/modules/dblog/dblog.module

/**
 * Implements hook_cron().
 *
 * Controls the size of the log table, paring it to 'dblog_row_limit' messages.
 */
function dblog_cron() {
  // Cleanup the watchdog table.
  $row_limit = \Drupal::config('dblog.settings')->get('row_limit');

  // For row limit n, get the wid of the nth row in descending wid order.
  // Counting the most recent n rows avoids issues with wid number sequences,
  // e.g. auto_increment value > 1 or rows deleted directly from the table.
  if ($row_limit > 0) {
    $connection = \Drupal::database();
    $min_row = $connection->select('watchdog', 'w')
      ->fields('w', ['wid'])
      ->orderBy('wid', 'DESC')
      ->range($row_limit - 1, 1)
      ->execute()->fetchField();

    // Delete all table entries older than the nth row, if nth row was found.
    if ($min_row) {
      $connection->delete('watchdog')
        ->condition('wid', $min_row, '<')
        ->execute();
    }
  }
}

only that your hook keeps the search entries or moves them to a different table.