I was curious about this as I may have the same need so I looked it up on the UPS website. I plugged in my city as the "ship from" and Tucson, AZ as the "ship to" because I know it's roughly 600 miles from here. I have a box which conveniently fits a two piece seat and is approx. 24"x18"x12" and the weight I had to guess at 10 lbs but the cost stays the same up to 32 lbs, Shipping from a UPS store or facility to a residential address would only be $37.34. That doesn't include insurance beyond $100 or any other extras like delivery confirmation, etc.
Distance really affects the shipping charges with UPS. That same box shipped from the west to east coast via Ground would be $86.34 and take 4 or 5 days.
Cool little tool:
https://wwwapps.ups.com/ctc/request?loc=en_US
Just reread the original post and realized it's a one piece seat in question. That would take a box approx. 38"x18"12" which bumps the cost up to $55.26. But, you could probably just avoid the box and wrap the ***** out of the seat with multiple layers of fine bubble wrap and stick a label on it and save some bucks. I know I've had large pieces of automotive hardware shipped to me like that.