Hi there,
My youngest child is 2 and I have a cat. I have four rescue dogs including two cocker spaniels
Most rescues will rehome certain dogs with young children and other pets. I foster dogs and cats for rescues and all of my fosters have (obviously) been suitable to go to homes with young children (providing the new home realise the time and effort involved) and other animals.
Sometimes the reason dogs cannot be homed with under 5's is that no history is known. I have occasionally taken in these dogs under strict conditions if they have been briefly child-tested, as have a few other parents of young children, BUT only because we now have enough experience to manage the situation and make careful introductions.
The problem is, there aren't enough fosterers, and more specifically, when fosterer's are available, there aren't enough experienced fosters who have children. Rescues can't hand pound dogs to foster homes with kids unless they know the person very well and have proof of their experience. This sadly means that many rescue dogs will never be tested around young children.
The best option in your case is to get in touch with rescues who do have foster homes where young kids reside, or look for a puppy, rescue or otherwise. My cockers came to me at 6 mths and 15 weeks..it is very possible to find pups in rescue as so many people buy a pup without realising what a lot of work it is and it ends up at the local rescue centre.
Obviously there are drawbacks though..for instance you will never know much about the background of most rescue pups, so the health and adult temperament of the dog can only be guessed.