How can I round the corners of the artboard to follow the rounded rectangle that I have covering it? (the end goal is export it as a PDF)
Most common image file formats, PDF included, are rectangular in shape. Everything is contained inside a rectangle of a certain size. Then, some formats will allow transparency to be used, in which case you can have rounded corners in the actual artwork, but not in the actual file itself, which will still have sharp, 90-degree corners.