After receiving the 429 error, do you know if they have any trouble re-publishing the posts to Discourse?
Can you get any details about how many posts are being published in succession? Do you know if the WordPress site is making any other API calls to Discourse other than the API calls that are being made to publish the posts?
Do you know if the WordPress site is using the new WordPress Block editor, or is it using the Classic editor to publish the posts?