“‘I’m fat,’ said Mrs Large.” If the first line of Jill Murphy’s A Piece of Cake, one of a series featuring a houseful of elephants, lacks the subtlety of, say: “All happy families are alike… ”, the impact is undeniable.

Some parents have, without getting much further, thrown the children’s book in the bin, or left their condemnation or warnings on various reviewing sites, such that the text’s alleged risks have now achieved national prominence. Under headlines mentioning “fat-shaming”, A Piece of Cake was itself targeted last week, as, it is occasionally argued, a purveyor of harmful messages about size and exercise and a potential contributor to eating disorders.

Adult readers have objected to a narrative in which, as horrified by her bulk as she is careless of food messages, Mrs Large puts her entire family on a diet, featuring “wobbly bits” and weighing scales. The publisher, Walker Books, summarises this episode in a popular series: “Biscuits and crisps are out, veggies and jogs are in! But when you’re a hungry elephant healthy eating and exercise is no piece of cake!”

Critics appear unmollified by the assurances of Mrs Large’s children that she is not fat, but “cuddly”, subsequently vindicated by the family’s harmonious reversion to former habits. The scales, by implication, are redundant.

Nor has it excused Murphy that her elephants are not, as typically happens to sweet-toothed individuals in classic children’s literature, pilloried for their appetite, regarded as obviously morally repulsive, seduced by their wretched gluttony into the clutches of evil, or depicted, unlike svelter characters, as doomed, such as Golding’s Piggy. Contrast the ultimate contentment of Murphy’s Large family with the fate of JK Rowling’s Dursleys or of Dahl’s celebrated, obviously asking-for-it victim: “Augustus Gloop, Augustus Gloop, / The great big greedy nincompoop!”

It may further inhibit infant diet ideation that the Larges are, unquestionably, elephants, not fellow preschoolers. A two- or three-year-old might feel no more connection with their weight issues than, for instance, she would with Obélix the Gaul’s boar habit, the young Babar and Simba’s bereavements, and Nutkin and Little Grey Rabbit’s traumatic sacrifices, to ruthless alpha males – in two venerated children’s classics – of their fluffy tails.

In Fierce Bad Rabbits, her intriguing new look at children’s picture books, Clare Pollard notices that the message of Little Grey Rabbit’s “servile” relationship with Squirrel and Hare is “that some people can be violated and exploited and this is their choice”.

As for the more conventional Large family: it’s enough for some parents, perhaps understandably, that the story, published in 1989, features words such as fat and diet, with the elephants recoiling, in flagrant defiance of Michelle Obama’s Let’s Move! campaign, from organised exercise. One UK study found that children are prejudiced, at age four, against fat people. “Pointless, and cruel. Avoid this one at all costs,” writes a Goodreads contributor, her instincts supported, if not by any evidence of picture book-linked disorders, by recent US guidance. A national reading programme has endorsed The Very Hungry Caterpillar for its exemplary nutritive contribution. Which is not much compensation for survivors of that deadliest of bedtime books, but better than nothing.

If it is time for a A Piece of Cake to go the way of Bunter, some dodgy Tintin and – at least for the Royal Mint – Enid Blyton, the focus on this book, out of all problematic children’s literature, may say more about the preoccupations of its adult readership than potential for harm. Is Murphy’s mixed message on empty calories more pernicious than, say, the glaring absence of male carers and preponderance of stay-at-home mothers in universally adulated picture books from Peepo! to The Tiger Who Came to Tea? More traumatic than the death of Mog?

Should fat elephants be discarded while children are schooled in the feudal structure – humbler rodents happily prostrate themselves before Lord Quercus Woodmouse of Old Oak Palace – at the heart of the popular, enchantingly illustrated Brambly Hedge series? “He often,” says the shameless Brambly website, “has to play a leading role in the many ceremonies that take place throughout the seasons.”

Lady Daisy enjoys – the Historic Houses Association could not have put it better – “welcoming the other Brambly Hedge mice into her home, and sharing with them the noble rooms, the ancient furniture, the paintings and the carvings”.

It is, of course, possible for the reader-aloud to challenge the hereditary entitlement of that smug bastard, Lord Woodmouse of Old Oak Palace, even as she indulges a young Brambly Hedge addict. Among the great pleasures of sharing picture books, before young readers depart to marinade themselves, alone, in fat-hating, moralising, colonial or otherwise unedifying messages from the past, is the opportunity for intervention, critical and otherwise.

“With picture books,” Pollard says, “there is a whole extra layer of adult interference, as the preliterate toddler cannot experience the story without a parent or carer mediating it for them (and often, as I do, changing vocabulary, skipping, unpacking, emphasising, doing the voices, pointing things out).”

Again, these interjections reflect parental priorities. While I tried, vainly, to foment local unrest against Lord Quercus Woodmouse, I tolerated the gendered servitude upon which The Lighthouse Keeper’s Lunch plainly depends, ignored homicidal speeding in Scarry’s Cars and Trucks and Things That Go. For every parent who diagnoses toxic influences in A Piece of Cake, another considers it classic Murphy, delicious escapism.

Either way, their children will soon have to evaluate for themselves Rowling’s fat Marge, fat Dudley and fat Vincent Crabbe; Dahl’s “enormously fat” Aunt Sponge. Unless, that is, this next-level fat-shaming is also ripe for suppression or replacement by more improving content: a revival, in fact, of the earliest, most doctrinaire intentions for children’s literature.

• Catherine Bennett is an Observer columnist