If you are thinking of any of the above calibers perhaps start with a large action, PGW Timberwolf or the big .416Rigby sized CZ. I suppose the Sako .338 Lapua can also be used depending on availability. This way you can rechamber/rebarrel and not have to buy a larger action. (Maybe have multiple bolts and barrels headspaced for switch-barrel usage?)
I don't really see a great advantage between the .338 Lapua and the .338RUM but, throw in the Lapua Improved and you have a big difference. The RUM's were Remington/Winchester's answer to the big Lapua's except instead of retooling and making a larger action they just made a new cartridge to fit the existing action. It all depends on availability of the components and parts, that should determine what to buy and use. (I can't goto the local gun store and buy a $55 box of .338 Lapua but I can get two boxes of .338RUM.) With that said I use a Armalite AR-30 and the .338 Lapua, I was not aware of the RUM until after I purchased the rifle... If I knew that the RUM existed I would have went with that and saved on ammo. (I wanted a large accurate heavy hitter that didn't use a belted case, .50BMG was too big soo... plus this was my step-up gun from a .22lr rimfire!)