I've had 2 custom shop rifle barrels; both were marked Custom Shop on the barrel. My custom shop handgun barrel, however, is not marked Custom Shop. You could call T/C and find out if Custom Shop barrels are always marked the same way.
For what it's worth; I don't think Custom Shop barrels are made to higher quality standards than the regular barrels it's just that you can get them in custom lengths, contours and chamberings. I could be wrong, but I don't think the 300 H&H Magnum has ever been a standard T/C chambering and therefore, if that's true, a 300 H&H Magnum barrel would be a Custom Shop barrel.