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I've rock-climbed in Peru (Huaraz π, Cusco, and Arequipa) and I've brought my own climbing shoes and harness (because they are quite light and also I cannot climb without my own shoes... maybe I'm a weirdo).
As for the rest, you can definitely rent it (you don't want to carry your rope and quick draws, it's just too heavy for your long hikes). I don't remember the price but it wasn't crazy expensive so that should be good!