You will get all sorts of recommendations and advice and they all should be equally weighed to make it as tough on you as possible. hehe,
I have read many say 250 gallons for each koi. The Nishikigoi Mondo book out of Japan recommends 2400 gallons per koi.
Here is what I think...
I think that will work. Koi produce much more waste than the typical comet goldfish. So, assuming each fish becomes 17 inches long totalling a potential of 85 inches of fish; I then double this since they're koi making it 170 inches of fish; this gives ya 1.4 inches per 10 gallons. HOWEVER, I am assuming an extremely light feeding schedule that is once every 2 days with food that is not extremely high in fish protiens. Typically, koi is an ornamental fish admired for its huge size and colors. So, a typical avid hobbyiest feeds the koi with a much higher protein quality fish food that promotes colors and fast growth to the point of the fish possibly reaching 36 inches long (if the fish's genetics allow it). So, if you were to take care of your koi to this extent, that is 36 inches long, and doubling the potential as I have mentioned, this would give you 3 inches of fish per 10 gallons; this indicates you definitely need a bio-filtration for your 1200 gallon pond and the veggie filter might be good enough depending on its size, how it is built, and what plants you use. I'm not a veggie filter expert, but I have read the inadequacies of them since they tend to be built too small with the wrong type of plants using the wrong type of plant medium.
If you were to put more fish in there that could possibly start totalling 4 inches per 10 gallons or higher, then I would definitely think about building your self a more efficient bio-filter or using multiple bio-filters. Save your money; don't buy the retail bio-filters. Bio-filters are far too easy of a DIY if you know how to simply operate a hole saw on an electrical drill.
Here's a picture of how big a koi can possibly become if fed well, environment is correct, and the fish's genetics allow it. Take notice that they are in a swimming pool type structure.