Solution:
You could borrow some code from the WC_Ajax::get_variation() method since that runs when you use select the attributes for a normal variable product.
$data_store = WC_Data_Store::load( 'product' );
$variation_id = $data_store->find_matching_product_variation( $variable_product, wp_unslash( $_POST ) );
Requires WooCommerce 3.0+