For books that you want to sell through Amazon, then yes, you should still use KDP Print. They offer a standard distribution in the United States and a European distribution option that you can enable, both for free, and books sold on Amazon in Europe will print and ship from Great Britain or other European printing centers so you can hit those areas of the world relatively easily.
Unfortunately, proofs and member orders placed through KDP Print still ship from the United States, so you can’t take advantage of the cheaper and shorter shipping options.
For your Ultimate Business Card types of books, if you live in Australia (for example) you can still use KDP Print for your own books at a net cost of an additional $2.50 per book if you purchase 50 or more (marginally less if you purchase more than that) and are willing to wait for a month or so before they arrive. In this case, it’s better to keep your book relatively simple and to proof your book digitally or by printing it on your own computer to catch as many errors as possible before having them printed rather than relying on a proof arriving, or to find a partner in the United States that would be willing to proof your book for you and to ship it to them. This could be very helpful when it comes to proofing your cover, as that would be the most difficult part to handle on your own without seeing what KDP Print does with the book.
If you live somewhere other than Australia, such as in Asia or Africa, then you can check the shipping rates and expected shipping times for your country at the KDP Print Book Shipping Rates page, which can be useful even if you live in the United States for calculating shipping costs for clients.
For the services that you offer to clients, you could still use KDP Print for everything and just handle the purchase of books for the client, and change the offer from “on demand” to requiring a purchase of 50 or more books at any given time with the expectation that there will be an additional time built in for receiving physical copies of their book. You could use that time to upsell them on digital options…
What I would recommend, though, is to find a local company that can create Print on Demand (POD) books and to work with them for your local copies you want on hand while still distributing the main book through KDP Print and Amazon.
I am not familiar with any international printing companies, but if you have experience with some then please email me and let me know your experiences; I’ll try to compile a list that people can reference.