David - if the koi are currently self-supporting on their food, it sounds like you have a pretty ideal setup for starting up a small aquaponics trial. If you can set up a basin/trough with some growing media to start some plants above the level of the pond as you mentioned in the OP, you could let the water run down through your current filter before returning to the pond until you are comfortable that the plant bed is purifying the water sufficiently.
Aeration through an airstone/airlift or just cascading through a fine rose when returning to the pond may also help your fish increase their size. Koi can reach sizes up to 2 feet in a pretty reasonable amount of time. A 12 foot long pond that is 4-5 feet deep at one end should have at least several hundred gallons. With good aeration, filtration and supplemental feed, quite a few pounds of some kind of fish can be raised per year.
There are several free documents that can be found by searching for low tech
aquaculture that describe ways of feeding omnivorous fish, such as carp. Everything from cut grass to manure is used to increase the nutrient density in the water to support fish growth.