Neat; something I know stuff about.
I don't have a trailer yet, but I know dog food like... never mind the simile; I'm just good with dog food.
The first thing I want to address is corn and other cereals in your dog's food. Back in history when feeding dogs first gained footing as a business, grains were incorporated into the meat to get today's "kibble", not because of a grain's nutritional value. Today we have flash-freezing technology that precludes the use of grains. Grains are still in most dog foods today not because they're still necessary but because they're cheaper. There may also be some "campfire talk" that dogs eating grain-based kibbles for ten generations should stay on such a diet, but I don't have a professional opinion on superstition.
Corn. "It isn't the first item on the list." Probably not; meat as the first item is the style these days. Keep reading the ingredients. How many grains are on the list in total? How many types of one grain? On one brand, I saw whole ground corn, whole wheat, wheat mill run, corn gluten, corn oil, rice, rice mill run, and more. Individual ingredients may be listed by volume, but each type of corn you can get away with adding gets its own individual listing. Ergo, when you see a meat as the first item, there may yet be three times as much grain by volume because of this exploit.
Corn. Corn is high in sugar, and dogs are more carnivore than grassavore. They certainly never grazed on wheat and corn stalks in the wild. Rice is fine for a filler, but the white version is as nutritionally null as Wonderbread. Wheat is high in sugar and corn is higher still. Further to the "why not corn" argument, both corn and wheat are considerably allergenic to dogs in an age of antibacterial, antiseptic everything stifling our immune systems to make us lethally vulnerable to peanut butter. Let your puppy go outside and get sick, or else he'll be on death's door the first time it sees a flea or tick as an adult.
So those are some basics about dog food from somebody that lives and breathes dog food. Now I'll share my tips for weight and value.
Non-grain dog kibbles can average a quarter of the size of grain kibbles without losing much weight. In my extensive studies of feeding my personal dogs a strict diet of whatever my customers left behind, I've learned that I need to feed an average of half the volume of non-grain kibble, and dog-dump is equivilently diminished in turn. It is however a considerable spike in protein even feeding less, and malabsorption and the dreaded loose stool demon can come to visit you, even while you're on the highway. Dog diets should not change during a trip.
A more dense kibble means feeding less by volume for the recommended amount means you can either buy smaller bags to last the same distance for saving space or you can have larger bags that add more total time between resupplies. Another tip is to avoid wet food because cans are huge, unfinished portions risk spoil and take up fridge space, and there's that water weight you're adding when you buy wet instead of dry. For anyone wondering grain or grain-free, look at a can of wet food. It should have "98% meat" written on it somewhere.
There should be dry or frozen food you're already bringing with you or buying. Rice, grindy beef, a teaspoon 'o yoghurt or cottage cheese, an egg, a sprinkle of flax seed dust. Flax is of course the highest concentration of essentiall goodies packed into one place by the way for anyone wanting to add that extra something to a minimalist diet of raw food or very nearly so. Yoghurt has the same value to a dog for its bacteria and egg is a shot of protein as long as you don't skim off the yummy yellow parts.
I'm sure there's more, but I don't spend so much time figuring out how to squeeze my dogs into my trips as what trips I can take with all my dogs.