You can backpack from the valley (up the Mist Trail, or the John Muir Trail) to a Half Dome (HD) "base camp" area called Little Yosemite Valley (LYV), where you would bivouac in a site overnight then hike up to and ascend HD the next day.
Bear canisters and special toilet arrangements are not required at LYV because there are fixed bear-proof food storage boxes and composting toilet facilities there.
As you have apparently deduced, attempting to hike from the valley to the top of HD in a single day is a strenuous endeavor (most who try it leave well before sunrise) and would be a very tough slog for young kids, I don't recommend it. I have climbed the cables to the top of HD twice recently, it is dangerous and not for the faint of heart. I would be fearful of children 8 & 9 trying it, but only you can judge their fortitude.
You need separate permits for camping in the backcountry and for ascending HD, and they are very tough to get. In fact, there are so many applicants that the permits are issued by random lottery. If you secure permits, they include a tent camping spot in the valley both the night before and the night after your backcountry permit dates, at the "backpacker's camp", along with parking for the duration.
I will say it is a lot of hassle getting it set up, but you will never forget it.