Enabling autocomplete for certain payment types only

Until now, I have enabled only payment with a credit card and, as my product is simply access to a paid web site, I am happy to autocomplete all paid orders using an action to that effect in my functions.php.
But now I wish to enable payment via SEPA direct debit, where there is a significant delay between placing the order and confirming that the customer has the necessary funds in her account. So I want to do one of two things, but am not sure how to achieve it:

Here is my current code, with the pseudo-code that I want to add:

function custom_woocommerce_auto_complete_order( $order_id ) { 
    if ( ! $order_id ) {
        return;
    }

    $order = wc_get_order( $order_id );
    if ($order->payment-is-NOT-by-SEPA) {
        $order->update_status( 'completed' );
    }
}

function custom_woocommerce_handle_SEPA_payment_confirmation( some_arguments ) { 
    if ( invalid_order ) {
        return;
    }

    $order = get_the_order_from ( one_of_the_arguments );
    if ($order->payment-is-by-SEPA AND $order->payment_succeeded) {
        $order->update_status( 'completed' );
    }
}

Any hints about how to achieve this would be welcome.