I have a couple of rifles chambered in 338 Kahn, which is an improved 338-378. The fireformed Kahn case has a little more capacity, but my load data should still work if you start at the low end and work up.
I just tried some of the Berger 300s using Retumbo.
102 grains 2895 fps
103 grains 2927
104 grains 2947
105 grains 2959
106 grains 2986
107 grains 3023
108 grains 3044
109 grains 3056
At 109 the bolt was stiff, so it was definatly OP. I have only done one round of testing and fired one round of each load. I really think 106, maybe 107 will be the max load in this rifle. This rifle is built on a Wby action with a 27" Lilja barrel.
I think with the stock 338-378 case you should be fine to start out at 102 grains of Retumbo and work up from there. I was using Federal 215M primers.
The load I have shot for several years is RL 25 with 250 grain Match Kings.
I get 3350 fps with 108 grains of RL 25 and Fed 215M primers. (30" Lilja barrel)
I would suggest you start out at around 102 or 103 and work up from there. This rifle has a 30" barrel and a Nesika action, so I don't think you will be able to get the same speeds because of the longer barrel and the fact the Nesika is a much stronger action than the WBY. Start at the low end. Your rifle may take the 108 grain load but I would certainly want to start much lower than that.