Short version...yes, 338WM is adequate for brown bear.
Longer version...
I've seen brown bears taken with calibers as small as 7mmRM with heavy for caliber premium bullets and proper shot placement...no problem.
Most would say the minimum would be a 30-06, but a .30 caliber magnum would be better if you have one.
The .325 (aka 8mm) bullets have been a favorite for bears for a long time and give a little more margin over the smaller ones. The 325WSM, 8mmRM, 8x57JS and 8x68S fill this role well.
Moving to the 338WM you are in the sweet spot for not too much recoil, having a light rifle and still plenty of power to do the job. The 340WBY, 338-06, 338-378WBY and 338RUM are also top choices in this caliber with varying levels of recoil.
Personally I'd be taking a 375H&H with some 300 grain Barnes TSX, because it's what I have and I know how it shoots.