I couldn't find a way to re-initiate django-autocomplete-ligth, as you asked, but found a workaround that solved the issue:

htmx.on('htmx:afterSettle', (e) => {
  const selElement = $('#id_of_must_hide_select_element')
  if (selElement) {
    selElement.addClass('select2-hidden-accessible')
  }
})