Both of my kids are very sensitive and easily troubled by scary and sad things in books. We've talked about how every good story has to have a conflict, but they still sometimes completely lose it over a character getting lost or separated from someone they love (Mango & Bambang) or mean/"rude" characters (Roald Dahl).
I've got a very sensitive 7 year old, who is now loving Enid Blyton's Naughtiest Girl series (she can't cope with Famous Five etc, too much peril!). She also enjoyed, when younger, Dirty Bertie and Horrid Henry, though as she was always thoroughly shocked by their antics I wasn't worried about the influence on her.
Paddington Bear stories are quite tame, Worst Witch. I find school-based stories, even if there is bitchiness, is easier for her to deal with than adventure peril, I guess because the setting's more familiar.
I’m the parent who wouldn’t read “Dogger”. My boy didn't mind the story, but I remember finding it really sad as a kid (even though it’s all OK in the end) and still don’t like it for that reason...
Roald Dahl has a nasty streak a mile wide, and while his sense of the grotesque appeals to most kids, it floats my boat much less now than it did when I was little.
I reckon ‘Swallows & Amazons’ and its sequels are a pretty rich vein. Grown-ups are only occasionally cross or disapproving, and the stories are gripping despite any threat or peril being largely imaginary (less so in Winter Holiday, to be fair, but it’s still one of the best).
The Land of Green Ginger, by Noel Langley - a very funny sequel to the Aladdin story, where Aladin’s son sets off to win the heart and hand of Princess Silver Bud. It includes a flying garden, a magician who has accidentally turned himself into a button-nosed tortoise, villains called Tin Tack Ping Fu and Rub Dub Ben Thud, and a donkey who sits on a pin.
Another vote for My Naughty Little Sister, and the Sophie stories (although i cried reading a few of those chapters .. DD said “just be brave, mummy”!) we also love Pippi Longstocking, Children of Noisy Village (& other Astrid Lindgren stories) and Mrs Pepperpot and another very old series of stories called The Cat Club. Also The Great Piratical Rumbustification, and others by Margaret Mahy
The little house books? Although you'd have to talk about the historical context as quite a lot of beating children for discipline. As said above Pippi Longstocking would be great - some mild peril but she can handle anything. I remember really liking 'the family from one end street' (Eve Garnett) but not sure if that is around still.