I work in the logistics industry and truly, whatever you ship you should ASSUME it's going to be tipped on every side. To go from point A to point B, *generally* you should assume that's going to be handled (loaded/unloaded) at least 6-7 times - from you loading onto local truck, unloaded at local terminal, loaded onto route truck, unloaded at next terminal, loaded onto local truck, delivered at local place.
All handled by people who, even if reasonable, are in a tremendous hurry all the time. Some - for example, if someone has to move your THING out of the way to unload his company's THING from the truck - doesn't give the faintest shit what happens to yours.
"Do not stack" and "No Stack", while some conscientious handlers will respect them, usually only cause one to hesitate before loading that 1000lb pallet atop your crate. If yours has a flat smooth top, it WILL get loaded on.
No, I wouldn't use fedex for such a shipment, either. I'd find a good local LTL firm where you could make special arrangements - the larger/more anonymous the freight company, the less anyone cares about your stuff: after all, if it's broken insurance will pay for it.