I thought rendering the fractals at a nice, big, resolution would make for more room to cut out my images from.

It did, but it also made the image I was placing all the cutout in 9000 pixels wide. -insert "IT'S OVER 9000!" joke here-

So, what do?

Rerender every image at a more manageable resolution.

(Fortunately it'll take half as long to render every image needed as it did to render ONE IMAGE at the bigger size.)