Set the search engine name for forms automatically enhanced by SearchWP Live Search

gistfile1.php

<?php

// only applies when SearchWP is active
function my_searchwp_live_search_get_search_form_engine() {
  return 'my_supplemental_engine';
}

add_filter( 'searchwp_live_search_get_search_form_engine', 'my_searchwp_live_search_get_search_form_engine' );