I think UPS and FedEx will charge more than the Postal Service, which is why I recommended USPS.
Actually no, Fedex is the cheapest of the 3 (for a long gun). Btw, I never priced DHL as a 4th option. When I had to ship K31s to C&R holders, UPS turned me around to their main pittsburgh office which is a bit of drive for me despite me having a C&R (they wouldn't handle ANY firearm unless at a main location). When I went to the main/local fedex office, they charged me approx $20 for each one (I had 2 to ship). They didn't ask me any questions, so I didn't tell any lies
USPS has new additional charges for non antique guns per my FFL and I can tell you from PA to Michigan with $300 insurance was nearly $35 from USPS but using fedex from PA to GA with $300 insurance was about $22. Big difference IMO.
As far as best choice, I would say this: if shipping an antique pistol or say a pistol or revolver with lead bullets (not loaded cartridges, just bullets) IE the gun with accessories which = additional weight, then USPS, due to the flat rate boxes. If its a long gun, I would go fed ex. Fedex doesn't ask you to declare what the item is unless its an international shipment. There are also no signs about weapons or ammo or anything prohibited. I'm fairly certain they don't care.
James K is right about legally being able to ship an antique firearm through USPS. You can ship most antique pistols, say 1911 or smaller for approx $15 with insurance with priority mail IE 3 day delivery under normal circumstance.
I buy and sell many things on ebay, and I started shipping guns to buyers as well. My info is first hand.