Provide SearchWP the necessary HTTP Basic Authentication credentials for the indexer to work in that environment

gistfile1.php

<?php

function my_searchwp_basic_auth_creds() {
	
	// NOTE: this needs to be your HTTP BASIC AUTH login
	//
	//                 *** NOT *** your WordPress login
	//
	//
	$credentials = array( 
		'username' => 'my_username', // the HTTP BASIC AUTH username
		'password' => 'my_password'  // the HTTP BASIC AUTH password
	);
	
	return $credentials;
}

add_filter( 'searchwp_basic_auth_creds', 'my_searchwp_basic_auth_creds' );