If your gunsmith tells you that rings with just a single screw per side isn't good enough for a 375 and that you need to go with a double screw set up, how about some mounts with 3 screws per side?
Those single screw rings have been holding scopes on rifles for years. The whole thing is to mount them properly and you will not have any problems. It is much like a friend of mine that called me years ago because he was having problems getting his rifle sighted in. We showed up at his home and I took a look at how he had mounted the scope which was loose enough that you could move it back and forth. I asked him if he had used any lock tite on the screws? He said what's lock tite? We got the scope tightened down and headed to the range where we got it sighted in. I told him that he was going to have to pull each screw one at a time and put some lock tite onto them and then resight the rifle in or at least take a few shots. To this day 40 years later I doubt that he has ever done that and still hunts with that rifle.
It was the rear base, not the ring.