It sounds like you have virtually the same chamber dimensions as me. I load to 4.108" with the 300gn Berger, single loaded and seated into the lands.
I know that you want to load from the magazine and I think that you should still be able to jump the bullets quite a bit and get good accuracy. You just need to play with your loads a bit.
On the positive side, with the way your chamber is cut, if you switch to Retumbo and single load out near the lands, you should get some velocities that might find amazing. Accuracy should also be quite good.
I have some load data that is applicable to my chamber that you might be interested in. Handloading the .338 Edge, a reloaders guide.