Tough one. As mentioned several times above, if you really want to use the .243 get the best 100 grain+ bullet you can and pick your shot. I would go with the 30-30, though. Not that it allows you to make a mistake, but that big ol' chunk of lead could lessen the anguish of not finding your animal after a marginal shot.
Whatever gun you choose, practice follow up shots! One shot after another after another. See if you can get 3 shots in a vital area sized target at your expected ranges offhand in under 10 seconds. 5 seconds is better and not impossible. Unless you get a perfect broadside shot at close range, you may be happy you practiced this. Elk are tough, and there are several guys with .300 mag and bigger that shoot and keep shooting until the elk is down for good.