I have not loaded the 338-378 but I have loaded the 338 Kahn which is the same thing but with a conventional 35 degree shoulder instead of the double radius Wby shoulder.
I use 105.0 gr H-50BMG in my Kahn. I would take about 12% off that to start in your Wby. so that would be around 92.0 gr to start with. Remember this is a custom rifle
so it may be different in your rifle.
H-50BMG is slower then you need for this class of round but it is very accurate and consistant in my rifle. Retumbo would probably work well also and US869 also works well but it more sensitive to temp changes. H-50BMG is hard to beat.
Just for your information, The Hodgdon sight lists US869 with this bullet in this chambering. Starting load is 108 gr and top load is 115 gr. May be wise to go with listed data and try US869. It will get you more velocity then H-50BMG, I have proven that in my Kahn as well, about 75 to 100 fps more velocity with top end loads in my rifle.