I'd go with the .243. I forget whether or not you said you reload. Anyhow, the Federal load with the Sierra 85-grain BTHP is as accurate as my handloads have always been with that bullet since long before the Federales figured it out.
If you reload, try the Hornady 70-grain or the blue-tip 55-grain, whichever brand those are.
Works good on Bambi, too, if you're a bit picky about the shot placement.