You can use several powders for this chambering. The 135 gr starting load works pretty well for all of them. For ball powders you can use AA8700, H-US869 and WC872 work well. You will get good load densities with these powders and they work very well for a static location and temps.
If you plan to shoot in a variaty of temps or elevations, you will see a quite noticable shift in performance using these ball powders. H-US869 was reported as being very stable with temp changes, that has sense been proven wrong with many experiments on this specific problem. Combine this with the extremity of the 338 Chey Tac with a 300 gr bullet and the velocity changes can be significant with temp changes as well as elevation changes.
That brings us to the stick powders. There are again several that would work but the two best are H-50BMG and VV 20N29. H-50BMG is about as good as it gets for this class of chambering. Stable, good load densities, consistant and accurate.
VV20N29 will get you basically the same performance as H-50BMG but with a bit more powder and a bit less pressure. I have seen some rifles have better ESs with 20N29 but that is not a given.
20N29 is a better powder for using bullets such as the 350 gr ULD RBBT where it will allow you to still keep load densities to the upper 90% range.
For bullets such as the 265 gr AT RBBT, H-50BMG is as good as it gets.
Again, the ball powders work very well but if you plan to shoot or hunt in a different area and temp then you develop your loads in, you better make adjustments accordingly.