Here's a thought; could we not easily raise BUGS to give us the bulk of the feed? This is what chickens eat in the wild to start with...
Earthworms. 30 small buckets (5 gallon) with drainholes set up on them, 500 worms in each. They reproduce in 90 days, low maintainence, easy to collect too all you need is a tarp on a sunny day. Feed one half bucket (250 worms, ish) every three days. Infinitely replenishing source of high protein, can eat just about any waste we have, even some from our chickens and provides us with great compost. Can be watered with rain water and fantastic garden benefits.
Crickets. Very easy to feed and maintain. Very easy. Can have a similar setup, just count out several dozen crickets each day for your penned chooks.
I think I would also keep and feed a fast-breeding fish. Doesn't take a big, appetizing, fish... Goldfish are fine, maybe mollies because they breed faster. Catfish are probably one of the best if you can manage them.
Offal from butchered animals for the packed vitamins and eggshells and crushed bones for calcium.
Green stuff can come easy from grass and weeds, but you could also dry whole grains and sprout them, especially in the winter.
In a societal collapse I just want people to think for a moment. We have seven MILLION cats and dogs going into shelters country-wide each year. That's with serious efforts to STOP over-breeding and with FREE spay/neuter facilities and 3-4 million being euthanized every year. We still get 7 million each year. What happens to those animals in the first year of societal collapse? Probably seven million cats and dogs out on the streets. The majority of those are breeds people were unprepared to care for. And the next year? And the next? You really think that with a few LGDs you will protect a free-ranging flock from a pack of half-wild feral pitbulls? Sorry. Not happening. Much like coyotes which can kill LGD, these dogs will team up in packs and tear your dog to shreds. The difference is they're not scared of people, unlike coyotes, which means they will NOT be deterred. Cats will sneak into small places to steal chicks and are almost as nimble as raccoons at opening doors, and are barely domestic even in their domestic form.
And what happens when some nitwit with a poorly managed exotics collection releases a dozen unfixed human-raised tigers and lions and bears (oh my) into the north American forests?
You may want to rethink your plans to simply let the hens loose. Before long we'll be making bomas like they do in Africa!