What you want is a fulfillment service. You have the printer print 1000 copies, then ship all of those copies to the fulfillment service's warehouse. Orders go through the fulfillment service's system and they ship directly to customers. They take a small cut from each purchase. This is standard practice, as far as I know.

https://unbound.com/how-it-works seems to suggest they handle this. Perhaps email them?

Maybe talk to the guys at bitmap books. They do Kickstarters for their retro gaming books, they really care about the print quality of their books and it always shows. The colours are always vibrant and the pixel art is crisp. They also get a fulfilment house to dispatch their books so they don't have to.

Maybe if you drop them a message they might be able to point you in the right direction