Here is a web site I have book mardked with the parent cases. The 300 Win, 300 Wby and 340 Wby are from the 300 H& H. The 338 Win is from the 458 Win case and where does the 458 come from?
you will notice a wildcat 334 OKH from the 300 H&H case which is 0.333 caliber then you go down the page and find that OKH did a 338 which is apparently so obscure that there is no listing for case dimension, but based upon what was done with the 334 it is probably too long for what you want.
There is a 338 A&H which is based upon a 404 Jeffery cut down which is short and fat as best I can tell and looks to have more case capacity than a necked up 300 win and is shorter. I would have to draw it out if it was me to actually understand the dimensions. My understanding of what you are looking for is a short fat fast 338 to fit your magazine.
I don't fool around with wildcats so you are getting really poor advice from me, but the website is interesting.