remove_product.php

<?php
$cart_update_string = "https://" . $store_domain . "/cart?cart=update&fcsid=" . $fcsid . "&output=json";

// loop our matches and remove
foreach ( $matched_products as $key ) {
    $item = $cart_contents['items'][$key];
    $item_id = $item['id'];

    // starts at 0, so increment.
    $index = $key+1;

    $cart_update_string .= "&{$index}:quantity=0&{$index}:id=$item_id";
}

$updated_cart  = wp_remote_get($cart_update_string);
$updated_cart_contents = json_decode( wp_remote_retrieve_body( $updated_cart ), true );