You can customize the checkout page using hooks.
To remove the payment options add the following code to your (child)theme’s functions.php:
remove_action( 'woocommerce_checkout_order_review', 'woocommerce_checkout_payment', 20 );
To add it after the order notes use this hook:
add_action( 'woocommerce_after_order_notes', 'woocommerce_checkout_payment', 20 );