Is the .338 Lapua improved the same relative to the original .338 Lapua in that it headspaces at the same place/location on the cartridge? Like the 30-06 Ackley Improved, 280 AI, 35 Whelen AI and others for example.
Use the cheapest/readily available bullet and a powder on the quicker side of the burn rates for the Lapua, will save some powder cost.
Load up mild rounds for some trigger time practice.
You can also do the cream of wheat method to save on components, look that up if you aren't familiar. Both these methods are under the guise that the improved chamber you have headspaces the cartridge in the same place as the parent cartridge.
My AI rigs will shoot parent cases very well. If I were chewing up all the expenses in the 338, I'd use some red dot and save the barrel and components for go time. If you don't have infinity rifle primers, a pistol will work for the COW / cotton loads.