I personally wouldn't hunt griz with anything less than a 338. The recoil is significant when you jump from a 300 to a 338 and that much again when you go to say a 338 ultra or 340 weatherby. I can't answer whether you can handle it or not. You'll just need to shoot one and see. Something to consider is the 375 H&H. Its a great bear gun with less recoil than the 340. Loaded with 250 grainers its deadly on elk, or with 300 grainers its a cape buffalo gun.