Flat Rate (free) boxes from the US Postal Service with signature required.

Before retiring, I owned an independent mail/packaging store for 28 years. Something like this, I would only ship UPS ground or Fed-X ground. Their accountability and tracking are FAR SUPERIOR to the USPS. 2 things - lay a plastic bag in the bottom and sides of the box and lay the quilt on top. Cardboard will find any piece of moisture anywhere and this provides a barrier. (I prefer not to put it in a bag because that is not breathable). And push it down to as small as possible. You will pay by size dimensions vs. weight. You can cut down any box by scoring across the inside, cutting down the corners to the score and folding (the scoring will allow the side to fold neatly). Be sure and put a card inside the box with the to/from info in case your label gets damaged. Coast to coast is 5 business days for both those services, so you can plan exactly when you want it delivered. These services may be a bit more, but not bad if you keep the size down and (I feel) it is so worth it. And, insurance goes by the $100 (first $100 is free) and it is very inexpensive. You can go online to pay and print a label and then drop it off anywhere that uses these services or go to a location. Reminder, if it is not an actual UPS Store or a Fed-X store, they will charge a surcharge. That is the reason many folks think these services are so much more expensive - they are paying the store markup. If it is a PO box address, you are out of luck. You can only use the Postal Service because the Post Office on the other end won't sign for a parcel.

I was able to get a queen in a large flat rate USPS box - I do not know how that compares to the other rates though.

USPS priority mail would be my choice. Also, I use space bags to condense the quilt down as small as possible. It doesn't harm the quilt at all, as it will spring back to it;s original bulk when opened.