Completely outdoor cat from scratch?

Indoor cats live an average of 16 years, whereas outdoor cats live an average of 4 years. Therefore, it is unethical to take an indoor cat and try to make it an outdoor one. The same goes for feral kittens still young enough to be socialized as indoor pets. On the other hand, it's very difficult (and potentially dangerous) to socialize older feral cats.

The simplest option is to hope that your new home has its own stray (not feral) cats or your new neighbors already have outdoor cats like your current home. If there is a significant feral cat population and your son is old enough, he may enjoy volunteering with a shelter or TNR group since that will give him controlled, limited exposure to cats, plus it's a valuable learning experience and service to the community.

The other option is to look into hypoallergenic breeds of cat. These breeds won't be found at a shelter, regardless of what they say. Note that "hypo-" means "less", not "zero", so you will need to visit the breeder to check your reaction to a specific kitten before you commit to buying. Many folks with allergies report no reaction, and others report a mild enough reaction that it is manageable with OTC medications and/or simple precautions like washing hands frequently.

My family has good experiences with adopting young feral cats, but admittedly we kept them in the house for a few weeks to make them associate our house with their new home.

Compared to getting an in-house cat, there are several differences to this approach.

1. You need a place to lock it up for a few days or weeks to get it used to living with you. A tool shed or something similar would be good, but the cat must not find a hole to squeeze through an run away. You must provide food, water and a litter box in this place and start socializing with your cat in there, too.
2. Kittens of a feral cat learn hunting from their mothers. Even if the kitten was separated from mom too young to have ever hunted a mouse on their own, they will remember and hunt too, as they get older. Expect a loving gift of a half-chewed mouse or bird on your door mat.
3. You need to offer a warm shelter during winter. Cats have fur, but their ears and faces are roughly as sensitive to low temperatures as humans are. A big carton or box filled with fluffy blankets or insulated with styrofoam on the outside is sufficient.
4. Due to living outside and hunting rodents, your cat will get parasites and gut worms. Save dates at regular intervals in your phone to remind you to freshen up all vaccines, dewormer and flea treatments, even if you don't see any evidence of parasites on your cat.
5. If your cat ever seriously scratches or bites you (maybe in a panic or for whatever reason), you must get the wound treated by a doctor. See this related question for more details. I was witness of how a small, but deep cat scratch resulted in such a severe inflammation that tissue from the hand had to be removed.

Apart from that, I see no arguments against having an outdoor-only cat. Make sure it was properly socialized with humans at a young age or it will always stay somewhat shy. If socialized properly it will trust you soon enough and - depending on its individual character - be a cuddly friend or a nice play partner.

Another solution would be a hypoallergenic cat breed (that doesn't cause allergies). Keep in mind, though, that these breeds produce less of the particles that cause allergies, but they still produce more than 0. So if you have bad reactions to very brief contact with normal cats, these breeds will still cause a reaction in you.

Maybe a combination of hypoallergenic outdoor-only cat would be the optimal solution for you.

My aunt adopted a young stray cat a while back. Since she was away during the week for work reasons, keeping him in the house was not an option, so she kept him outside (house with a garden in a rural area). She had a garden shed where he could get in for shelter. Her partner built an electronically controlled feeding station (also in the shed): he wore a token around his neck which would activate the feeding station when he approached it (after an activation it would be locked for a certain period of time).

The cat was 5–6 years old when he disappeared (she suspects catnapping, as a few cats in the village disappeared that same summer). By that time my aunt was already retired and at home during the week, so there would not have been any more obstacles to letting him in.

So I think it is doable if you provide food and shelter outside the house, as well as regular vet care, and if the cat is used to being outdoors. Of course, they are subject to all the dangers of being outside, as with any domestic cat that is free to roam outdoors. Such is the price of freedom. On the other hand, if you adopt a young stray cat, he will most likely have a better life than he would otherwise.