For the record, the .45-70 MR BFR really isn't bad for recoil. The low pressure old BP round is about in the same league as the .454 cassul, so honestly, the BFR doesn't make sense to me. Why carry a single action THAT STINKING BIG when you can carry a double action Taurus Raging Bull or Ruger Super Redhawk?

But, I digress... The poster wasn't interested in handguns, nor in handgun cartridges...

My personal recommendation for lightweight/easy handling, and HARD HITTING bear medicine is a Marlin 1895 Stainless Guide Gun in .45-70. You could also go the .450marlin route and save yourself time in reloading. Slap on a $75 set of XS ghost ring sights, and feed her half a dozen 325grn Hornady Leverevolution FTX's and you'll have a fast handling <7lb rifle that's able to knock down any animal on the planet.

Semi-autos rifles in a self defense situation are a liability in my book. If that thing jams, you're done. A Marlin levergun isn't going to jam on you, and she'll run plenty fast.

Or, my second choice would be a youth stocked 18.5-20" rem 870. The disadvantage here is that it doesn't have much OTHER practical use (the Marlin will make a fine hunting weapon, a short barrel shotty, not so much). The good news, is that a mouthful of 00buckshot is mighty persuasive even to the most persistent of critters. I suppose you could pick up a spare barrel and swap it over for some wing shooting when you don't need it for covering your tail.

I know you said no handguns, or handgun cartridges, but a Ruger Deerfield Carbine (.44mag), or Marlin 1894 in .44mag would also make fantastic bear defense. Plenty of stopping power, very light and fast handling.

Just for fun, if you WERE considering a handgun, something on the lines of a Ruger SRH or Taurus Tracker would just fit inside your price range. Hard to argue that a handgun is quicker on the draw than a rifle or shotty, and having the opportunity to operate it one handed (while a bear has you on your back, chewing on your OTHER hand) is a pretty big advantage.