Your initial post was the catalyst to decide and plan on our own family visit to Fuji San, but we are chained to our kid’s school holiday dates which means late July to the end of August - avoiding the 12th to the 20th Aug and weekends for our climb.
For us it is more of a pilgrimage, up past the forests it becomes a place of the spirits, not of the living. Saying that, MrLapLap is keen not to go anywhere near the forests at the foot of Mt Fuji towards the North West (Aokigahara).
We seem to be decided not to go watch the sunrise from the summit. We’ll stay the night somewhere high to acclimatise ourselves, see the sunrise somewhere quiet on the trail and, hopefully, avoid the worst of the conga crawl where the Subashiri and Yoshida trails converge at the 8th Station.

What has been really good about taking our cue from you has been the opportunity to grab some winter outdoor gear in the Sales. Much easier and a heck of a lot cheaper to get the important items now than it will be nearer the summer.
My kid now has a half price 3 in 1 shell and puffer jacket combo from Polarn O Pyret (feels poignant buying a child’s size for the last time) and whilst trawling online through TKMaxx to find what else there might be for her I unexpectedly came across a solitary 2022 Goldwin insulated jacket, in my size, at less than a third of the original price. I had to stand around in the cold for over a hour last night (0C and windy) and then use the tube. I didn’t get cold outside, didn’t overheat inside, it’s a really good jacket and weighs less than 1kg. Am delighted! So thank you, Petec, for prompting our purchase (MrLapLap found himself some things on the Mammut website). It’s going to be weird taking this stuff along in August though.

Boots purchase for the kid will have to wait, no way to know how much she might grow in the next few months.
I’m wondering whether I should risk using my old boots which I was planning to retire and letting them get completely knackered on a last hurrah, or scuffing the outside and shale peppering the lining of some new boots with the demands of the climb/descent.

Sun protection is another concern. MrLapLap just got the first sun hat I’ve ever known him to have. I unexpectedly found a pale pink Hijab for £3 from Under Armour which looks meant for summer sports, not worn one before but it’s thin and moisture-wicking and looks ideal to wear along with a cap. Am not keen on putting on loads of sun cream when I can’t shower or bathe it away and have to sleep in the grease.

Which reminds me, we’re also looking out for thin cotton sleeping bag liners to take and use in the huts. The bedding must collect loads of sun cream and other oils, am guessing a liner would mitigate this inevitability.