The most efficient way is to avoid loading big images like that. Are you sure you can't just make the background out of smaller instanced objects?
Or, would I be better exporting the background image as one oversized power-of-two image (in this case, 1296x1296)?
No, don't do this; some (most?) newer graphics cards can handle textures that aren't square. Whenever possible, try to make square power of two textures, but if you absolutely need a big rectangular image, just keep the texture as small as possible, and try to keep height and width the smallest power of two possible. Don't add blank space just for the hell of it; You lose performance for both the old and newer graphics cards that way.
would I be better off cutting up each backdrop into a sequence of power-of-two tiles, say 256x256, and importing them as seperate objects?
You save some VRAM this way, especially if your texture is a really irregular shape.
But like I said, always try to use small objects to build the background, if possible.