If he is only having the reaction to underwear then it's most likely not latex.
Latex is in lots of other things that you would have noticed by now. They use latex in elastic, boots, gloves (Any time a nurse touched him while wearing gloves he would have broken out), swim suits (If you touched him while wearing yours or your husband either he would have broken out). Tie some elastic around his wrist, loosely of course, see if he reacts to it there too.
My thoughts are the laundry soap. Under wear fits tight and rubs the skin constantly. This is the first time he has had anything fit this tight against any part of his skin. Even shorts move and hang away from the body from time to time during the day.
Try using all different laundry products (soap, fabric softener, and dryer sheets) just for a few loads and see if it makes it worse, better, or the same. That way you can prove or eliminate the cleaning process as the culprit. If you notice a difference then you need to narrow it down and see if it's the soap, or the bleach, or the fabric softener, or dryer sheets. Each item can be the cause.
Try different pairs/styles of underwear. Are you putting them on him right out of the package? He might be allergic to the sizing products they treat the fabric with. Almost every new garment has some sort of substance on it to make it look pressed and wrinkle free and when it's washed it looks totally different. If it is indeed the underwear waistband then I would think about making him some. It would be like making little shorts and you could use 100% cotton, pre-wash the uncut fabric in the hottest water you will ever do laundry in, that way it will never shrink any more that the first wash and dry. Then dry it at the hottest setting you'll ever use too. I use warm water for everything except light/whites that I use bleach in. So if I were making underwear with 100% cotton in a very light or white fabric I would wash it in hot and dry on high. You can use a drawstring, buttons, snaps, very soft sew-in Velcro circles, any number of closure options are out there. Think about how they held up their underwear before the invention of elastic.
Good luck in your adventure of finding what is causing this rash.