It will bring your bear down but not as quick as an LFN started at no faster than 1600 fps, and use it on anything up to big brown bear and moose. The length of a 350 gr is nice for getting accuracy, but that much weight isn't needed once it gets to the animal, even for moose. Also, 2000 fps with 350 gr is going to make the rifle come back pretty hard, which is fine if you don't mind.
Story time? I loaded a 300 gr LFN to 1800 fps in the Marlin 44 mag, and took one deer with it. The wound was bigger than I like, so the deer was able to bound around in a complete circle before expiring, whereas at velocites from 1450 to 1600 fps he would have collapsed in his tracks. Also of interest. The bullet hit the off side shoulder joint and pushed it out thru the hide, ripping a 2 1/2 to 3 inch hole in the skin, with the bone shards of coarse. I had to load the 300 gr to 1800 fps to get stability at long range, and it is a sweat shooting rifle, though the stock end seems to bite a bit. I selected that bullet and load because I can hit deer etc easily out to 400 yards with it, once I get the trajectory 'felt out'. A better choice for bullet would have been a 280 gr loaded to 1600 fps, a speed where it stabilizes nicely.
You've got a big cartridge because you thought you needed it, so will perhaps think I'm blowing smoke. Ask me any questions you have and I'll relieve you on them. But don't throw out your nice rifle. Just load it so it's easy to shoot and enjoy it!