I use the next to largest cooler at Walmart (measure and I think you need 31 in length and 14 or 15 width and height.
Coleman Xtreme 120 Quart Cooler - Walmart.com
A complete trans and the cooler weighs 144 pounds and FedEx will ship up to 150. I get a reduce rate and it is about $110, retail might be about 130 bucks. I think this is the best way to ship. I usually screw the lid closed with some sheetrock screws