Depending on the size, shape, and type of packing material you need to protect it, the price could vary wildly, but by all means use your own box. Flat rate boxes are only cost-effective if you have something that may or may not break the usual straight-up priority weight limit. I've been shipping USPS for 12 years and many a time had the postal clerks help me save shipping money by choosing my own box instead of flat rate based on the weight. Size does matter, though.
Use the calculators on both sites to estimate shipping, but I can almost guarantee you that UPS will cost more. They, and FedEx, specialize in overnight/express shipping over the tried and true postal system, so their rates are significantly higher overall. Even UPS ground will likely cost more than USPS. But I second Cyanna's suggestion of using the shipping calculators on the websites. Input how much you expect it to weigh WITH BOX, and what the dimensions are, and they can tell you what it should cost.
Founder and Vice-President, Madison Area Costuming Society, a chapter of the International Costumer's Guild
Strange Land Costuming - www.strangelandcostumes.com
Director, SF&F Masquerade
- COSTUMECON34 - Madison, Wisconsin - May 6-9, 2016 - www.costumecon34.com