Keep your total dimension (length + girth) less than 84" and you won't pay any extra. Go to USPS and learn the rules because clerks often do not. I prefer to ship from home with USPS click and ship. It's cheaper and the postman picks it up from your house.
I have shipped bows and bow blanks in all kinds of packages but the safest, cheapest, and easiest is 4" PVC for longbows and 6" PVC for recurves. You might have to partially brace the recurves to get them to fit. Some just don't and you'll have to make a box.