Should I bring my birds to storm shelter?

The area I live in gets dangerous weather around this time of year and in the spring, and the biggest threat is tornados. I would not think twice about bringing my two budgies with me if we took shelter in a basement but we have an underground storm shelter in the backyard and I was wondering if I could get some opinions on whether I should take them with me or clear a spot in my closet and leave them in there in case of bad weather. My concern with taking them to the underground shelter is that it's typically raining very hard and quite possibly hailing whenever we do go down to the shelter and it's a very small space for four people and a cage so I would have to put them in a box (with holes in it of course) to carry them out in and keep them in until we come back to the house. We have never had a tornado actually do damage in my neighborhood but you never know. Would it be a little too traumatic to go to the shelter or would it be in their best interest?