If there is not a moldy smell I would use and try to get the smell out afterwards.
Batting inside bags may not smell.
Washing batting may come apart and ruin your washing machine.
I have been making quilts for about 50 years. I don't have a basement, I do keep a damp rid hanging in my fabric closet.
I mainly use high loft polyester batting and love it. I quilt on a 9" throat machine.

My sincere condolences on the loss of your parents. It must be very hard to deal with. I hope you find comfort in making memory quilts.

Let me add my vote for baking soda, if mildew is the issue. While researching refrigerator issues recently, I learned that baking soda is the antidote to mildew.

Just my $ .02

We had a body odor problem in a man's suit at our house and dry cleaning did not remove it. I began a search for anything that would remove it. Fabreze worked sort of but what worked the best is a product by 501 cleaner. I sprayed it on the underarm area inside and out. It took several applications but the odor finally disappeared. There are these odor removing sprays in several product lines and already in a spray bottle. Good luck with your problem.

this was mentioned earlier, but if the batting was stored in the bags it comes in as when it is purchased, very good chance that the batting itself will have no odor. I too have successfully washed both poly and cotton battings on gentle cycle and dried in dryer with no problems whatsoever. It also helps to get rid of the wrinkles. Unless the product has mildewed, there is nothing that couldn't be washed out and used.

I would try the "week outside in the freezing cold". If you do try washing, use the "delicates" or "hand wash" cycle with a little vinegar added to chlorine bleach (1:4) to kill any mildew. It is safe to premix, but don't try to save the mixture because the bleach will break down. (That's why it doesn't come that way.)