This question comes up all the time. Everyone has their opinion on this, but from what you have written, I would say that you would do well with the .243. I'll probably start an argument, but forget the neck shot! The high lung shot works best; has the best margin for "error" and often drops year in their tracks. As for proof of the .243 effectiveness and shot placement, here is a reply I wrote about two years ago to the same question which you have asked:
During 20 years of professional taxidermy, the biggest bull elk I ever mounted was for a 16 year old kid who used a .243 with Barnes bullets. That's not a caliber that's recommmended for elk, but shows what you could do with the right bullet and good bullet placement even with a relatively small caliber. Here's a picture of that mount. By the way, the big bull dropped dead in its tracks to a heart shot!