This huge egg here, is it sort of tethered to the ovary, filling up the coelomic cavity, and then exiting out of the infundibulum, or, where is it in the bird exactly? What "space" does it occupy?
In birds, eggs mature post-fertilization in the magnum and "shell gland", sometimes called a uterus.
I can't find anything to suggest a kiwi is any different, despite their large egg size. An exception is that kiwi have two functional ovaries (yet the join in one functional oviduct and the rest of the reproductive system is only present once) (Ballard & Cheek, 2016). Given this source mentions this specific difference, it seems unlikely they would have omitted any other important differences.
This course page may be useful for bird reproductive anatomy more generally: http://people.eku.edu/ritchisong/avianreproduction.html
Ballard, B., & Cheek, R. (Eds.). (2016). Exotic animal medicine for the veterinary technician. John Wiley & Sons.