A domesticated cat, especially one who has been an indoor cat their whole life, does not belong outdoors. I believe that if you cannot provide her the shelter she deserves, yes - you should rehome Star.

Being outdoors exposes her to not only predators such as wildlife (coyotes) but cars and weather extremes in addition to disease even if she has been vaccinated.

Good luck in your decision. I know it can't be an easy one and must be weighing heavily on your mind.