Oh, I’m sorry, I misinterpreted your query – that is not the button you want.
The actual settings button in the search input can’t be changed via templating, but there is still hope.
Try adding this code via the Code Snippets plugin or to the functions.php file in your theme/child theme directory – make sure to have a full server back-up first for safety. For more details you can check the safe coding guidelines.
add_filter( 'asp_shortcode_output', 'asp_change_search_button_class', 10, 2 );
function asp_change_search_button_class( $output, $id ) {
return str_replace("<div class='prosettings", "<div class='sok-installningar-" . $id . " prosettings", $output);
}