In case anybody has the same problem/question and you happen to come across this post:
I contacted Woo support regarding the question and asked whether Woocommerce Order Status Control plugin would enable me to do what I described. After receiving a useless AI-genereted answer and asking for a human to answer me (and waiting for a day), I actually got a solution from the support:

1. Install Code Snippets plugin if you haven't: https://wordpress.org/plugins/code-snippets/
2. Copy/paste this code into the .PHP section:
add_filter( 'woocommerce_payment_complete_order_status', 'wc_set_addon_orders_to_processing', 10, 3 );

function wc_set_addon_orders_to_processing( $status, $order_id = 0, $order = false ) {

	if ( 'completed' !== $status ) {
		return $status;
	}

	if ( ! $order && ! $order_id ) {
		return $status;
	}

	$order = $order ? $order : wc_get_order( $order_id );

	if ( $order ) {
		$items = $order->get_items();

		foreach ( $items as $item ) {
			if ( ! empty( $item->get_meta( '_pao_ids' ) ) ) {
				$status = 'processing';
				break;
			}
		}
	}

	return $status;
}

Seems to be working for me!