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.
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 unless you purchase your books directly from Amazon European sites.
That means you will either pay more for your books that you want to have on hand by purchasing directly from Amazon or purchasing expedited shipping, or you’ll have to wait (in some cases a lot) longer for your books to arrive, and depending on where you live you’ll have to wait longer and pay more.
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.
Depending upon your time constraints and budget, it may make sense 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.