The biggest limitation with the 338RCM is powder capacity. So part of your question is what bullets and bullet weights are you planning to shoot?
For example, if going with something like a 185gnTTSX, you might need to go faster than H4350. Perhaps try H4895. If that is not available, I would try Varget, that you listed above.
For heavier bullets, e.g. 225gnTTSX or 250gn, you might find that H4350 can produce a decent load, but you will probably want more space and should consider StaBall6.5. Hodgdon lists 59.8 gn max load, and uncompressed!, in the 338-06 with a 250gn Hornady for 2572fps. [The 338-RCM has about 2 grains more capacity than the 338-06, but the RCM has a short neck and loses its nominal advantage with medium to longer bullets, so that it is very similar to the 338-06, sometimes getting by on a grain or so less powder than a 338-06 load with longer/heavier bullets.]
Another space-saving alternative would be CFE223. It might be best to use a magnum primer with CFE223 and would require that you re-zero and/or re-test between summer and winter loads. That can often be accommodated by adding or subtracting a grain of powder.