Well, never hunted bear, so take it with a grain of salt, but the 243 ought to do it, from what I have read. I would AVOID shoulder shots with the 243. Also, in my opinion, don't get too hung up on the KE numbers. Trust me the 30-30 has a better reputation with big game than the 243 does. When kept under 150yds the 30-30 is a solid elk and moose rifle. Those 170gr bullets open a big hole.
Yours obviously will have no issues. I have heard of the 325 BT's opening very early. Personally I'd rather a 45-70 bullet not open than too early. I hunt with the 300gr TSX in mine, but only cause it's the only affordable, and legal bullet for this stupid state. It also consistently prints a 3 shot clover leaf at 100yds.
Not cutting your choices, I think they will get er done. I have heard the same thing about black bears, they are easy to kill, when both lungs are taken out. My 243 is loaded with 95gr partitions. I like deep penetrating bullets. I figure a smaller hole through both lungs beats one lung splattered any time.
Good luck on your hunt.
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