Can I set a custom “alt” attribute with woocommerce_get_product_thumbnail()?

You can’t with woocommerce_get_product_thumbnail(), but this will give you the exact same result:

global $product;

$image_size = apply_filters( 'single_product_archive_thumbnail_size', 'woocommerce_thumbnail' );
$custom_alt="Put your custom alt text here.";

return $product ? $product->get_image( $image_size, [ 'alt' => $custom_alt ] ) : '';