Not to sound like a big A-hole but.........................

If you are wanting to hunt grizzlies then forget about the recoil. You're just gonna have to live with it. With biger bullets comes biger recoil. There are a whole bunch of ways to mitigate recoil while at the bench and while afield though.

Having said that the 340 Weatherby would not be on my list of cartridges and while I would prefer something larger than the .300 I would not feel "undergunned" with one.

A good round to consider would be a 35 Whelen. It will launch a 250 grain bullet at about 2500+ fps. That's a pretty big slug making very respectable energy levels and will give great penetration with very manageable recoil.