Elwood Epps chambered quite a few rifles to 338-303 in the late 50s and into the early 70s. His suggested load was the Speer 275 gr bullet ahead of a case full of 4831. The velocity was around 2100 fps. Remind you of anything?
I never owned one myself, but a lot of guys had their old 303s rebored and chambered for it, and I never heard any complaints.
Pacific Tool and Guage lists the .338-303 as an off-the-shelf reamer.
Ted