Yes meat and mayo can go bad. Use an ice pack in an insulated lunch bag. They're reusable. I'm saving my ice pack and insulated cooler bag that I used for expressed breastmilk for when my little one is in school. An old beverage bottle with a little air in it can be frozen and re- frozen and doesn't have all of the chemicals that a commercial ice pack would have. It might leak eventually, though, so maybe tuck an cloth napkin in the lunch bag just in case?