We've done it with a 3 year old and a 9 month old. Instead of stuffing the trailer to the gills with kid junk, we leave everything at home (except a minimal diaper kit). That leaves the back of our trailer (the Burley Bee) free for carrying food. If you need to augment this arrangement, add two canvas bags across your handlebars. This way you can carry a good deal of weight in front and back. As for Costco, the only way I've found to cope with the weight is to shop more often, as it costs next to nothing to bike to the grocery. The alternative? Wait until one of them goes to school, then stack their weight in food beside kid #2. Could be a long wait to hit the store!
Another alternative? Go on the weekend with your mate, also on his bicycle (extra hitch already installed), and make them haul the trailer on the trip there or the trip home. There are several other permutations here involving trail-a-bikes, bike seats for infants and toddlers, and tandems.