I have a friend who has hunted elk in the area of Paonia, Colorado for over 20 years running. He reports that there are a good number of women and others with an aversion to recoil that hunt elk every year with 243s.
There is no reason that a 243 is not acceptable with proper bullet choice and shots limited to broadside and not too far. 200 yards should be fine with the right bullet.
If it were me, I'd prefer to step up to a 25-06 or 7mm-08 and consider the 7mm-08 to be ideal.
But the 243 will certainly work for a responsible, knowledgeable hunter.
Still happily answering to the call-sign Peetza.
You do not HAVE a soul. You ARE a soul. You HAVE a body.
He is no fool who gives what he can not keep to gain what he can not lose.
-Jim Eliott, paraphrasing Philip Henry.