I have been parting out some cars and shipping some larger stuff. Check out Fedex.com and UPS.com for shipping quotes. First measure your bumper cover. Then you can play with the measurements on the website to get the price down. On an exhaust I just shipped yesterday, I found the line between it costing $40 and $80. Just 1-inch smaller box made the difference.
Also, have the measurement of your box when you go into the store. A lot of times they'll take your word for it, so tell them what the box measures, and they might not measure it themselves. That boxed exhaust was so close to the $80 size, I was worried that it would measure larger depending on how they measured it. But I just told them what size it was, and they were happy with that!
I have also found that FedEx has been cheaper than UPS for what I have shipped, and also FedEx will ship stuff larger than the post office will even handle.
Hope this helps!
Richard