If you do your job the .243 WSSM will kill a black bear. I've only shot one and I chose a .30-06 for the job loaded with 200 grain Nosler Partitions. Bullet selection will be the key here I'd go with the heaviest grain bullet available in Nosler Partition/Accubond, Swift Scirocco, Barnes TSX, or Speer TBBC. I'm guessing you will be shooting from a stand and the range will be close so you will want a bullet that will hold together at your velocity.
I really don't have too much experience with premium bullets other than Partitions and all I can say is that they work. Try and knock your bear down with the first shot, I aimed for a high shoulder on mine and dropped him in his tracks. This means you will need to break bone so a premium bullet is a must.
The only problem I see with the .243 caliber on bears is that they don't leave a very big exit wound. I found that my bear wasn't hard to kill but, I really don't want to have to track anything that is wounded with large teeth and claws through the forest. If you have another caliber that is larger or could afford to purchase one, you might consider it.