Re: Which Neck Sizer to use
Dunno what would be the very best. The best product I've used for case sizing is Imperial Case Wax. I could be surprised, but I don't expect this process of setting the shoulder back to be all that difficult.
And taking the necks down will be very simple provided no individual step-down process is too large. If the case neck shoulder collapses, you'll know it's time to add an extra step to the necking down process. Any pistol resizing die that you have on hand might make the neck resizing process more gentle. A 38 Special or 357 Magnum case resizing die might be an option for the first case neck reduction step. That would only be 0.018" step down in the first process. That's a very common die. If you don't own one, perhaps a friend does? Less is always best. I'd try that and see if works well.
Then move on to the 338 RCM die for another 0.019" reduction. That would turn the one-step process into a milder two-step process from .375 to .338. When I have to neck down wildcats from parent cartridges, I use anything I already have available (meaning no additional cost) to size down in smaller steps.
Another means for a two-step process between .338 and .308 could be a 325 WSM (.323") full length resizing die, if you have no-cost access to one. I do believe you'll be able to go from .375 to .338 and from .338 to .308 with just one-step each, and wouldn't spend money to break those down into two-step reductions. But if you have a no cost option to size down through a four-step process, like .375 to .357 to .338 to .323 to .308, then that's what I always try to do.
I have a relative that owns a gun shop and I have been known to borrow used reloading dies (pistol or rifle) from whatever is available there in the shop to progressively size down case necks. Which of course is helpful. But sometimes I borrow various dies from a friend, since they're only for one-time use.