I think the OP is looking for a provider, like Ingram Spark, that does POD hardcovers but with an API so the ordering process can be automated. That is, a customer can go to the OP's website, order a hardcover book, and the OP's website sends that order information to the POD book printer's website to order and ship the book. The OP's website can already do this for other merchandise because companies like printful have integration options where your website can communicate with theirs. I believe the OP is looking for the same thing for POD hardcover books.
If you're looking to automate your IngramSpark orders, I’d recommend checking out my company Demon (demondms.com). It integrates with your e-commerce store and IngramSpark so your customers can place orders, and Demon handles the print-on-demand ordering automatically—no need to manually fill orders.
It works great with Shopify, so you won’t have to change much if this is what you're using already. It also supports platforms like Squarespace, WooCommerce, and others, giving you flexibility depending on where your site is hosted. It handles international orders (like selecting the printing region, and it even does the two factor automatically if you set it up to) and there's no limit to the number of orders.
There's good documentation, it ingests your products from IngramSpark automatically when you set up the integration, and for individual publishers with at most 10 POD products it's just $19.99 a month. It's not free, but it also takes away a lot of manual work. It also handles things that other systems don't, like placing POD orders across multiple different distributors.
They haven't played ball with any integrations, and I added this feature over the last year because of the closing of well-loved Small Press Distribution, which forced lots of small presses into tons of manual labor they couldn't afford to be doing. It does it by acting like a user - Demon has 3 dedicated servers that actually go out and do the work of placing the orders, getting the products, etc, 24/7. It has lots of fail-safes, and is currently processing the orders for 2 medium-sized publishers every day.
The documentation for Demon's IngramSpark, WooCommerce, Shopify, and Squarespace integrations are all easy to find from the link below.
https://docs.demondms.com/core/integrations/distributors/ingram-sparkDemon is a tool primarily used by small- to medium-sized publishers, and it has been around for about 5 years now. Of course this doesn't change the rest of the discussion around KDP vs Lulu vs Ingram vs another, but for those using Ingram, it's something.