RE: 243 for bear
I killed the biggest black bear i ever shot with my .243 win. He was walking side ways at about 40 yards to me, and i used 100 grain Nosler Partition!! One shot, and it was all over. BTW, this bear weighed just over 500 pounds live weight!! (yes i weighed him)
In my opinion the 6mm's are too small for big deer and the bears because there's no margin for error. I'd choose something a little bigger if possible.
I've shot quite a few foxes, and coyotes with both my .243 win, and my also my 240 Weatherby. Of course i wasn't trying for "perfect" pelts, so i mostly used hunting loads. They are rather hard on pelts!!