I would try to find a Weatherby Mark 5 because of magazine length,(Can be increased to over 4")
bolt face can be opened to fit the 338 Lapua or if you find a 370, 416 or a 460 it only has to be opened .007 and the extractor works just like it comes.
These action are very strong and I build all rifles with cartridges larger than a standard magnum
bolt face using the mark 5 action. They also look and function great.
You can also buy a flat top bullet follower and center feed mag box for it from Weatherby to feed the larger cartridges smoothly.
They are not easy to find at a good price but if you hard enough they are out there for $600.00
to $850.00in very good shape.
JE nailed it! Also look at total cost including rail base, any truing, tactical bolt knob (if desired) and the possibility of replacing the weak and breakable Wby extractor for these large cases. Then compare this to the Stiller TAC 338 custom action. I know from experience of doing both which way I would go.