Once upon a time I was given a pile of 375 H&H brass and it formed into 338 with just a run through the 338 die and a big trim. I almost mentioned this as an option many of us might have to resort to, but looking at shootingbot for in stock 375 shows it’s just as hard to get as 338.
300 win mag brass is readily available though - I’ve heard of guys using a tapered expander, annealing the neck, and making 338 in one pass through the die, but personally I’ve never tried it.
If I was making it from something else, I’d start with 7mm Rem Mag brass. Should be able to run it into a 338 die and be done. Neck will be a little shorter than standard at first.