This is your golden opportunity to organize your stash so you can find anything at any time.
I have fabric in my shed (which is built of wood and has a good floor and roof and walls), which does get hot in the summer and cold in the winter. I was also concerned, but it's been there a couple years and has not been harmed. It is in cardboard Banker's boxes, bought from Staples, and each box has a clearly labeled, tightly sealed white trash bag around it in case of pests. My fabric is by color on the color wheel, that is, yellow, yellow-orange, orange, etc. I turn the box on it's side and load the fabric so that I see the folded crease of the fabric and therefore see all fabrics when I take the lid off. For my fabric, it is on shelves that are labeled with the color as per the color wheel. So if I need a blue-violet, for instance, I go to the blue-violet area of the shelves and find maybe three boxes. I check them and usually find what I need.
Since you are concerned about your family's reactions, you could do this sorting and packing when they are not around. You might mix the boxes in with other boxes if you feel the need to hide them. Then have a plan for where to put them after the carpeting. You could also find someone with a huge stash and show them, and compared to that stash, yours would be small.
You could put a big tent up in your back yard and use it either for storage of general household stuff or make a "room" for your family through the carpeting.