There may be other problems with the follower and feed rails. In Rem 700 actions, you can not load RUM cases (which are basicallty same diameter as WSM cases) into a 25-06 rifle because the rails are slightly too tight.
I would sell the Browning and get a Weatherby Sub MOA Vanguard in a WSM. A much better rifle in IMO.
I re-barreled a rem 700 30/06 to 7mm rem mag. The gunsmith opened up the existing bolt to accept the larger cartridge. I don't know if the browning bolt can be opened up like a remington can, check with your gunsmith.
As montanarifleman said the feed rails will probably need to be opened up. You might need a magnum magazine box as well.
You could pay someone to do the conversion that entails altering the bolt face and feed rails mag box, etc. and hope that it feeds and functions ok when all is done. After paying the cost, you would end up with a long action rifle chambered for a short action cartridge and about 100 fps better velocity at the same chamber pressures.
For the same or less money, you could buy a new rifle already chambered for the 300 WSM round if you want to run one.