If you are only choosing from these two cals, 308.
The Lapua will kick like a mule and cost several times more per bang IF you could get components.
Like the RUM, there will be many Rems in the Lapua with almost new barrels.
Personally, I don't like shooting a 308 in a light rifle all day long. I prefer the 223 which does the same thing for even less cost and recoil. My 1 mile capable 223 reloads cost LESS then a 308 Matchking bullet alone.
Both are effective to at least the mile. The 338 does drift a whole lot less which makes shooting in the wind easier. But then driving a 223/308 in the wind is a very rewarding challenge.
A 6mm, 6.5 would do just as well as the 338. A 7RM will beat it.
Unless you need the horsepower, there are so many 'better' choices like the 223, 6BR, 260R, 7-08/7RM, etc.
If you really want a boomer (been there and built several LR monsters), take the cost of the Rem and apply it to a custom. You will get a much better shooting rifle that can extract all that performance WAAAAAYYYY better.
At which point, I would suggest you study the RUM. Does the same thing and will be considerably less money and headache to build.
Jerry