Nimrod, if you want a good blood trail, you most definitely want an exit wound. The .243 Winchester will consistently fully penetrate an Elk if using a bullet such as the Barnes X. Of course there are exceptions to the rule such as if you hit both shoulder balls or some other fluke circumstance that rarely ever occurs. As far as exit wound size, increasing 1/2 or even 1 MM is not going to make a tremendous difference in the size of the exit wound. Double it due to expansion, and its still not a huge difference. You get your huge exit wounds from hyper velocity bullets driving bone and from the shock wave exiting. My 7Rum exits with a wound the size of a base ball, but a 7-08 will not leave a wound even close to that size. If your daughter likes everything about the 06 except the recoil, consider having a muzzle brake put on an 06. I believe you will find that a 7-08 or a 6.5 Creedmoor will leave an exit wound slightly smaller than the .243 due to their lower velocity.
I have yet to see an elk(non archery) that required an exit wound at all, no less the size of a baseball to drop. this year alone my younger brother took an elk with my 300 weatherby magnum, exit wound about the size of a quarter, dropped 10 feet from where it was shot. my brother inlaw also killed an elk with a 45/70, didn't exit... 7mm-08 is more than capable of killing elk with proper placement and good bullet construction.
with a 7mm-08 you can get a 160gr projectile traveling at nearly 2700 fps. compare that to a 243 where your absolute best is a 105 gr bullet traveling 2900fps, believe it or not those 55 grains make a pretty big difference. also compare to 7mm rem mag where 160 grains max out around 3100 FPS, 4000 feet per second difference doesn't matter much inside of 300 yards.