Well I guess I will be the first to say what hasn't already been said.

Most black bear hunting is done over bait and out of a tree stand which gives you time to pick a shot at a calm animal. If this is how you will be hunting black bear then there is no reason at all that the .243 that you already have shouldn't be used. A 100 grain Nosler partition bullet will take down any black bear that you will encounter.

Now if you are going to be spot and stalk hunting bears then I would opt for a larger big bore lever action round such as the .444 Marlin or 450 Marlin or a bolt action cartridge like a 30-06 or 35 Whelen just because they hit harder. Which is what I would want when hunting bear from the ground.