Any D/R can't be inexpensive.
Buy a decent hard case and pack it inside securely. Then cover the hardcase with cardboard like you'd wrap a present.
Cardboard sheets can be had from places like Office-Max type stores if you can't find a used packing box of some sort to slice apart, lay flat and use.
Label the inside hardcase as well in case they do get separated.

Yes the hard case can be slip out, stolen and the box continued on it's way. So can a DR in a cardboard box be slip out, stolen and the empty cardboard box continued on it's way.
All you can do is try to prevent shipping/handling damage while in transit.

Getting in to which carrier is best/worst is simple.
Ask a dozen people who have shipped guns and which carrier they've never had a problem with.
..They all come w/risk, everyone has a favorite till they have problems with them. Till then, they're great.