Interesting thread. I don't see the Mormon link at all; in fact, I think Lewis was implying that the Scrubbs family were atheists, as part of their 'modern' (negative connotations for Lewis) living. Lewis would probably have seen this lack of religious belief as part of their 'faddist' nature, which explains the underwear:
Douglas Gresham (Lewis' stepson) said:
"... At the time there was a fad for string underwear, it was crocheted out of quite coarse cotton thread and was in a net form. It did feel rather weird to wear but it was very warm... Harold and Alberta were simply faddists."
The 'faddist' angle would seem a reasonable conclusion to draw from the following paragraph:
The Voyage of the Dawn Treader said:
There was a boy called Eustace Clarence Scrubb, and he almost deserved it. His parents called him Eustace Clarence and masters called him Scrubb. I can't tell you how his friends spoke to him, for he had none. He didn't call his Father and Mother "Father" and "Mother", but Harold and Alberta. They were very up-to-date and advanced people. They were vegetarians, non-smokers and tee-totallers and wore a special kind of underclothes...
Additionally, Eustace is a materialist, which lends credence to the idea that he was probably an atheist (pretty much confirmed later on - see
The Silver Chair quotes below). He prefers fact to fiction, proof to imagination:
The Voyage of the Dawn Treader said:
...Eustace Clarence liked animals, especially beetles, if they were dead and pinned on a card. He liked books if they were books of information and had picutres of grain elevators or of fat foreign children doing exercises in model schools.
Lewis took this further. He might have seen families like the Scrubbs as anti-traditionalist, 'faddist', taking up with 'modern' ideas he disapproved of. He reserves particular contempt for Eustace's school, the significantly named 'Experiment House'. He has some very dated views on education, believing that corporal punishment was an important element of school life, and also finding the idea that girls and boys could
possibly learn anything in a mixed environment ridiculous:
The Voyage of the Dawn Treader said:
...at each word he gave Eustace a blow with the side of his rapier, which was thin, fine dwarf-tempered steel and as supple and effective as a birch rod. Eustace (of course) was at a school where they didn't have corporal punishment, so the sensation was quite new to him...
The Silver Chair said:
...I shall say as little as possible about Jill's school, which is not a pleasant subject. It was "Co-educational," a school for both boys and girls, what used to be called a "mixed" school; some said it was not nearly so mixed as the minds of the people who ran it.
Finally, Lewis casts his 'Experiment House' as a secular institution:
The Silver Chair said:
"I'm not," said Eustace. "I swear I'm not. I swear by - by everything."
(When I was at school one would have said, "I swear by the Bible." But Bibles were not encouraged at Experiment House.)...
The Silver Chair said:
..."Son of Adam and Daughter of Eve, hey?" said the Dwarf. But people at Experiment House haven't heard of Adam and Eve, so Jill and Eustace couldn't answer this...
This would presumably be yet another aspect of Experiment House's foolishly 'modern' ideas. Lewis implies that ideally, a school should segregrate genders, beat children when they were disobedient, and indoctrinate them with Christian texts. In the context of a man born in 1898, this is an unsurprising perspective; it sounds horrendous to me, but then I was born 83 years later.