Many Tears do not have blanket policy against rehoming with small children. They were going to let me foster with a three year old in the house, but I am too far away and have no reliable transport. Start there.
You can also search on Dogs Blog for dogs that are suitable with children. Dogs Trust are a little more strict but will rehome the right dog with small children.
Smaller, independant rescues tend to consider on a case to case basis with no blanket policies about rehoming with children.
I can tell the truth about my dog and make him out to be brilliant, if I were to advertise on him pre-loved i.e "3 year old terrier x. Good with other dogs and cats. Lives with children. Walks well on his lead. Energetic and friendly. Very loving and enjoys his cuddles"
All of that is true. What I have not mentioned is that he has a very expensive skin condition and has a history of aggression. He does enjoy his cuddles and he is very loving towards me. No-one else. He is very possesive of his space and this can lead to aggressive behaviour. He has snapped at the children he lives with before. But if I mentioned any of that in a pre-loved ad, no-one would buy him.
People lie. They lie to experienced rescuers, (this is why all decent rescues vet and assess the dogs themselves) they will certainly lie to the general public. No decent, loving dog owner worth their salt would rehome their dog via pre-loved. It would be better for the dog if it did end up in a shelter, since dog fighters and puppy farmers scour pre-loved and the like looking for free and cheap dogs to abuse.
here are all the dogs on Dogs Blog that can live with children