Let her handle and shoot both and choose for herself.
The .243 is fine for many types of game but if elk are on the menu I'd start her with the .243 and work her up to the 7mm-08. I would also consider getting a a youth or compact stock if it would fit her better.
Coyote Hunter - NRA Patriot Life, NRA Whittington Center Life, GOA, DAD - and I VOTE!
No, I'm not a Ruger bigot - just an unabashed fan of their revolvers, M77's and #1's.
This discussion does show the merits of handloading, as the recoil impulse from a 120gr 7mm-08 at 2,600 with 31gr of 4198 is 8.9 ft/lbs, while a garden variety 100gr .243 factory load at 2,900 is making 10 ft/lbs. They are effectively the same on the shoulder. A 120gr/2,850fps load using something like Varget, RL15, or 4895 is going to be 12.3 ft/lbs by comparison, while a well-built 100gr .243 handload that is bumping 3,100fps with something like RL26 is going to turn up 11.7 ft/lbs. Again, the recoil is almost the same. This all assumes 7.5lb rifles.
My daughter is shooting Remington 700MT rifle in 7mm-08 and we have great results with 110gr ttsx and RL15. She has shot 2 small buck and they both went strait down in there tracks....
I used to load 120 nbt's and a starting load of H 4895 for my nephew. He killed at least 3 bucks with that load. I think it ran about 2800 fps and shot moa from his 700 youth rifle. Even as he grew bigger he still wanted me to load that load for him because he said he had faith in it.
For nearly 30 years my go-to 7mm-08 load has been 43 grains of WW-748 under a 139 grain Hornady Spire point. I will soon be trying 110 and 120 grain TSXs in my 7mm-08. If they shoot as good as my go-to load, I will switch. Less recoil, better killing power - what's not to like?
My kids are shrimps and definitely recoil averse. To get them going I used trail boss in a couple youth model .243 that would put a 58 GR VMax to about the same POI at 100 yards as an 85 TSX. Trailboss and a light bullet makes for almost no recoil and they can plink at rocks, milk jugs, etc for a long time without getting sick of it or flinchy.
When it is time to hunt I switch to the 85 TSX. Has worked well for me.
When they are big enough to go to the elk woods I switch them to a Ruger Frontier 7mm-08. The heavy barrel on the Frontier helps tame recoil/muzzle jump and all have made the transition smoothly.