To the OP, yes a 338 will do the job but the only thing about Browning is that you are stuck with the barrel you've got because Browning permanently fits their barrels to the action.
Until recently I also thought there were issues if a person wanted to rebarrel an A-bolt. It isn't true. There was a recent thread here where many owners posted about their rebarreled rifles.
To the OP, yes a 338 will do the job but the only thing about Browning is that you are stuck with the barrel you've got because Browning permanently fits their barrels to the action.