I think planning for something that is impossible to accomplish is a waste of time. A fit person with a lot of experience walking long distances that has some means of resupply along the way could walk that kind of distance.

No one could do so carrying a dog and a kid and all the gear and supplies they would need.

Thinking about how one might do so is not a terrible exercise as it will become very clear that it is just not something that is reasonable. It is like people who seem to think that retirement planning is about winning the lottery. There are some people who win the lottery, but as a plan for retirement it just plain sucks.

The problem with the Mormon human powered carts is not that it can't be done, but that the people who did it routinely walked long distances already because it was their primary means of transportation. If you are not walking 5 miles or more several times a week on an ongoing basis, pulling the cart that carries a kid and a dog and gear and supplies is not reasonable either.

It is not like you can go buy such a cart anyway. The people who used these kind of carts had a huge advantage over a single person in that they almost always went out in groups. If a cart broke down they had help fixing it. If they needed to go hunting or foraging for food, there were people who were better at that to do it. They had food preservation skills that made it possible to more or less live off the land, although I kind of suspect that like many who went west, a lot of the meat they ate went with them on the hoof.

I sort of like the bike and trailer idea, but it has its own issues. It is a lot more reasonable to bike 400 miles hauling a dog and kid and gear and supplies in a trailer than it is to walk carrying them on your back. because you can make a lot better time on a bike, you might well be able to carry all the supplies you need. Having done some 20+ mile biking when I was much younger, I can attest to the fact that even though I was on nice paved paths the whole way, I was pooped after 20 miles, and pretty sore the next day. No way I was going to do it again until I made 400 miles. And pulling a trailer behind is a LOT more work. Even so, it is a lot more reasonable than the 400 mile hike if the OP is willing to get a sturdy bike and trailer and start doing 10 miles a day with a 100 pound load in the trailer.

The problem is that the OP never said anything about a bike. The OP planned to walk 400 miles, which just is not possible given the constraints she mentioned. She probably could not get 400 yards carrying the dog, kid, and gear on her back, even forgetting the supplies needed for a 400 mile walk.