Hello, Our family hiked the TMB last late June, early July and brought our tent. We wild camped one night out of necessity and as long as your hidden and out of sight no one will bother you. Our plan was to stay in two or three mountain huts but unfortunately they were all full so book ahead if you go that route.
We stayed one night in a hotel in Courmayuer, Italy and booked that the night prior while we were at a campsite on the hike. We found that all the campsites have showers, food, and WiFi so that was pretty plush from what we are normally used to out backpacking.
As far as moving your pack daily, we saw a lot of people doing that, but if you travel light and stay in huts and hotels you should be fine carrying a day pack.
I know my pack was about 45 lbs, my spouse had 35 lbs, and our 11 year old daughter had 15 lbs., and that was all gear, (tent, sleeping bags, thermarest, stove, fuel, food, first aid, clothes, etc.) We hiked it counter clockwise in 11 days, 10 nights and that felt like a good time frame to not feel too rushed.
Wonderful experience and very popular. The section through Italy was the most scenic and best trail conditions.
Have fun!