Talked to my postal folks today and they said the "Special Handling" was actually "Special Handling-Fragile" and was/is typically used to ship items like live chicks, animals, etc. or other really fragile things (like art work, pottery, etc). It does get special handling but it is not insured at all. You can buy insurance but it get expensive. I would not think that the USPS would allow a firearm to be shipped that way but who knows.
They told me, that for firearms, the best way possible (not necessarily the cheapest) is Priority Mail with insurance. It gets also special handling in that anyone who touches the package has to sign for it and they are the first packages that are loaded/unloaded. They did say that yes, an insurance claim may take a while but the people who investigate the claims are like a PD Internal Affairs office, they are independent of the postal service per se, they just want to find out what happened, and who caused the loss/damage. The postmaster said they do a full investigation to make sure the USPS is not being scammed and they want to be fair to the shipper and receiver.
I haven't have any problems in over 20 years of shipping with USPS so I'll probably stick with them.
This has been interesting because FFLs and non-FFLs alike always want to be able to ship firearms the quickest.