Using 338 Lap brass as the parent cartridge will give you thicker walls than using real 416 Rigby brass, so it is a plus in my book.
In reality, pick a cartridge and go with it. When you handload with a reasonable case, you can generally find a powder that will get you to the performance level you desire, especially if you have a long barrel. Take a look at what the Palma shooters do with the 308.
Between the RUM and the A.X. I'd be tempted to get the AX solely because of the availability of Lapua brass as the basis for the round. No one I know of ever got looser groups switching to Lapua.
However, at the ranges you are talking about shooting, the difference in a quarter MOA at one hundred yards is irrelevant once the wind picks up. But it never hurts to have a more accurate load.
Machine guns are awesome until you have to carry one.