The 375 H&H operates at lower pressures than the 458 Winchester.

Insofar as shooting the action loose, it isn't pressure but backthrust you have to consider. Calculate the backthrust from a 12 gauge shot shell and you'll quit worrying about your rifle.

The double rifle is perfectly capable of handling just as much pressure as a bolt action rifle. The weak point is extraction. The DR just doesn't have the mechanical advantage in extraction that the bolt action has. In the early smokeless days they experienced difficult extraction and kept the pressure low for that reason. The difficult extraction eventually proved to be a matter of pitted chambers and weak brass. The problem went away with the advent of clean chambers and better brass.

That said, there is no reason to run high pressures in a double since you are free to make the case as long as you please. The bolt action is forced to high pressures because you are limited in the size of the case. The only real reason to chamber a DR in 375 H&H or 458 Winchester is the readily available ammo.