I don't have a 338/416 (yet) but I've had 338 Lapua and 338 RUM and one can figure to get 2775 to 2900 fps out of these with a 27.5 to 30 inch barrel and the Sierra 300.

These two cases are about the same in case volume and are essentially shortened 338/416s.

As for using a 505 Gibbs, I recently had a fella tell me Prairie GunWorks had something in 416 on a 505 Gibbs case and the 408 Cheyenne is using a case derived originally from 505 Gibbs dimensions. I don't know how well a 338/505 Gibbs would work, probably way overbore and beyond efficient use.