Nero,
One more thought.
Given that the barrels are both modern steel and much stronger than the original, there IS a way to shoot the .375 or .405 without beating up the action and breechblock. (Not that the ~140 year old stock might like .405’s)

“Back when”, P.O. Ackley experimented with backthrust on the bolt/locking lugs when firing a cartridge. (He used Winchester lever actions).

Mr. Ackley found that if a moderate pressure cartridge case (say less than 40,000 psi) and the chamber walls were clean and grease/oil free then upon firing the case would grip the chamber wall and not impart any force or backpressure to the bolt face.
He even went so far as to remove the locking lugs from the test rifles and when fired they would not even move the unsupported bolt back.
The only thing that backed up to the breechface was the primer.......the solid head of the cartridge case contained the pressure with no stretch or separation.

So if you were to thoroughly degrease the cartridge and chamber before shooting it likely would reduce or eliminate any potential stress on that old action. All the pressure is contained by the chamber walls and thick web of the case head.

There is the esoteric bit of thought for the day!

- Mike