And you never lube a case for inserting into a chamber, It will create way too much rearward bolt thrust, Especially on a case like a .300 H&H, It has a very tapered case compared to a 300 Win Mag, But putting any type of lubricant in a chamber is bad juju, It's a well known fact.
Try a go-gauge for a belted magnum, One gauge fits all belted magnums except those based on the .378 Weatherby. If a go gauge fits, Then you have other problems. Are your cases formed from .375 H&H? If they are, Then your necks are way too thick and you need to turn the necks or ream them, So the problem could be the necks are too tight.
Edit to add: There is one other Belted magnum that a standard belted gauge wont fit, And that's the .224 Weatherby Magnum, Kind of strange in that it has no parent cartridge, It's an oddball for sure!