How to wrap select list in custom html?

Question

I have a select list created by views exposed filters.

When I debug the templates used. I have the exposed filter template, then the select list is in a seperate template.

I tried just putting my html in the exposed filter around {{form}}. This put the html around the whole form, including labels, other containers etc, which didn't work for my situation.

So I am trying to target the select list directly using this answer here

function HOOK_theme_suggestions_select_alter(array &$suggestions, array $variables) {
  $routeObject = \Drupal::routeMatch()->getRouteObject();
  $view_id = $routeObject->getDefault('view_id');
  $display_id = $routeObject->getDefault('display_id');

  if (!empty($view_id)) {
    $suggestions[] = 'selectlist__' . $view_id;

    if (!empty($display_id)) {
      $suggestions[] = 'selectlist__' . $view_id . '_' . $display_id;
    }
  }
}

On first load of view it works. When I update the view using ajax the twig template is no longer loaded.

I check the template suggestions and it suggests the same twig template for before using ajax and after. I am at a loss on how I can edit the select list for a specific view whilst maintaining ajax use.