As with any other structural question the answer is the bigger the better. Normal thread for the 338 Lapua is 3/4-24. A 5/8 seems a little small though it will work. The biggest issue is whether you're looking for really high accuracy or just a blaster grade setup. With many barrels and steel in general removing material allows the internal stresses to move things. Run a flat piece of stock through a mill and take 50 thou off one side and it will curve toward that side....unless its been stress relieved before milling. The same is true of barrels. Cut rifled barrels tend to have less stress build up because they're removing material evenly inside though it is possible even for them to have stress buildups from the making of the barstock used to make the blank. Button rifling definitely induces stress into the steel which needs to be removed by the stress relief heat treat process.
Some will tell you that fluting can make the barrel bend and I've seen it happen so know its possible. With better barrel makers who stress relieve after button rifling its no so much of a problem. As for cutting the muzzle down for threads this is applicable as well. Removing material from the diameter of the barrel outside can allow the inside to swell. In watching a large number of barrels go through the shop that have been threaded after building I see some that have copper fouling in the bore but not at the muzzle because the bore has swelled ever so slightly where the threading was done. Because of that it makes sense to remove the least amount of material you can and use the largest thread that will work. I thread all my barrels up to .358 cal 3/4-24 for that reason and have seen no problems with swelling bores.
Check the bore at the muzzle end of your barrels and see if you notice any change in the copper fouling there compared to the area not under the threaded portion. You might be surprised at what you see. Opening up the can to fit the barrel is frequently not a problem though with some manufacturers it can be impossible because of thread linings or space constraints. I'd open up the thread in the can or brake before cutting the barrel down any smaller.
Frank