Allegra,
Great question.
As long as the pack does not have a metal frame, and I suspect it does not,  you should wash it in the tub, as you suggested. If it has a hydration system, remove it first. Then fill the tub with slightly warm water. It’s best to use a mild, powder detergent, such as Forever New. Gently move the pack back and forth in the water for a few seconds, and then let it sit for three to five minutes. Rinse with slightly warm water and then hang it to dry.
Best of luck,
Aly