A dog's breed will put certain things in the bag and it's own individual personality will add yet further. But that is like a travelling bag of potential and to a greater extent what makes the dog is you.
Some Springers can be frenetic and scatty - channelled properly you get a quick, busy dog that works all day; homed inappropriately you get a dog that is neurotic or destructive.
Labs are great dogs but they are also boisterous.
Cockers are jaunty and meant to be busy - show cockers are still cockers, although less frenetic they are equally prone to quickly amusing themselves with destructive behaviours. It's all about you.
It sounds like you and your DS have worked so hard to overcome his fear. If you get a puppy he will be bitten, chased - can he cope with that? Many children without his previous fear have become scared or wary of puppies that do this. Of course the first thing you do is start training but in the meantime is this an appropriate situation for him? (not snarky, literally just asking).
Puppies are also hard work, like reverting back to having a new baby, they are not easy blank slates.
My personal opinion, that you haven't asked for, is the perfect dog for your family is going to be a schoolmaster/mistress - these dogs are older, usually out of their teen months althought you can get very steady puppies. I would really recommend looking to rescue. If you have a breed in mind there are myriad breed rescues nationwide who will work exceptionally hard to match you up with the correct dog. Breeders may also have older dogs - pups that have been run on to assess their potential but are not suitable for showing or breeding - not problem dogs.
Otherwise I would think hard about your needs as a family and your circumstances - show strains of gundogs may have been calmed slightly but many people work and show together, they're still dogs made to be out in a field all day. We are back from a show today, last weekend they were all out competing as gundogs. Speak to a few breeders, they should know each pupies personality like the back of their hand so be prepared for them to say the one you fell in love with when you laid eyes on it is not the best for your needs. Also be prepared for a good breeder to have concerns about selling a puppy to you - we would have to meet you both and think long and hard but, being brutally honest, from a starting point of no.