Build one of each and see which one handles pressure better.I read David or someone state why the normal tenon diameter will be just fine. The XC case is the same diameter but about .45" longer than the 338LM. More pressure is applied over a larger chamber area, which is approximately proportional to the additional powder capacity. Volume of cylinder=approx. extra case volume=.293 x .293 x 3.1416 x 0.5L=0.135 cubic inch. Additional case surface area=0.585 x 3.1416 x 0.5L=0.919 sq.in. This is a simplified equation since the taper and shoulder angle comes into play but what you will find is the surface to volume ratio for the extra cartridge length is roughly the same as the ratio for the case itself before adding length.
I think there was more worry with using existing 338LM barrels which when reamed, extended past where the taper began. I ordered a 33XC Bartlein ModBB in Heavy Varmint with the full diameter 5-inch section prior to taper. I wouldn't expect any issues with that even with 37XC. Do the math: .375-.338/2=0.0185 chamber wall difference. I can't imagine slightly more than 1/64th difference in thickness would matter, again, factoring the additional pressure spread out over a larger area.
Guys on the long range hunting forum are running Deviant Tactical Actions (1.0625” X 16 T.P.I.) with 37XC and having no issues. Chamber wall of .344 vs .362 for .338. Basically a hair more than a 64th.
I’ve build cheytac diameter stuff in a savage shanked barrel. It won’t run the pressure a big shanked action will. Very noticeable