I have a 2.75X Burris Scout Scope on mine, but I had a custom quarter rib made for the purpose of mounting the intermediate eye relief scope.
I used a M-70 .375 with a 2.5-8X Leupold in Africa, and found it to be a convenient size, and even at 8X had acceptable eye relief. Set at low power, the rifle could be brought into action very quickly, but it would certainly be the highest magnification I would want on a .375.
The only problem with a very short scope like a 1.5-5X, is that you might need extension rings to bridge the magnum length action. When it comes to mounting a conventional scope, particularly a variable with a long ocular bell, I would use an extension rings anyway so that the scope could be mounted in such a way that the ocular lens did not extend past the cocking piece of the rifle.
I would consider a scope mount from Brockman that features a pop up, ghost ring, which is locked down while the scope is in place. Considering what a .375 is called to do, I think iron sights are indispensable.