Hi Dana, the 162gr ELD-X, SST or a bullet of similar weight will have the most energy downrange in this cartridge using RL16 or H4350. At 500 yds you can still have about 2,000 fps and 1,500 ft/lbs of energy with these bullets. This is with a 24" barrel so you will lose a bit of speed and energy with a shorter barrel on a compact rifle.
Hornady's 150gr ELD-X, 154gr SST or Interbond also with RL16 or H4350 is another option, you'll can still have 1,500 ft-lbs at 400+ yds depending on the load.
Lighter bullets will have more speed out the barrel but will lose it faster and won't have the energy that the heavier bullets will.
Dont know of your daughter is recoil sensitive, but heavier bullets will have a bit more recoil however there isn't much difference in this cartridge.
I have 2 7-08's that I have loaded for extensively but neither has that short of a barrel (22"/26"). Of the loads for the shorter barrel that worked well, Varget and a 140 Accubond might be a good place to start. I loaded 41.5 grs at a OAL of 2.8" in Lapua brass and a BR-2 primer. This load was safe in my rifle, take the customary approach, reduce and work up slowly. I have since switched to Staball 6.5 for the same rifle.
Elk! Smaller caliber/cartridge! Young shooter, with recoil a potential issue! The Barnes 120 TTSX pushed hard, would be on my “very short” list of candidates! Assuming, that the ranges be kept within reliable velocities for good expansion.....which for most should be a non-issue! memtb
If you are shooting at closer ramges say 300 yds or less a lighter bullet will do fine, but a heavier bullet will work better, it will have more energy. The heavier bullet will also work best at longer ranges where the lighter ones just don't have the energy, especially for an elk.
The heavier and higher BC bullets will outrun the faster lighter bullets any day. For an Elk sized animal more is better and it won't limit your ranges like a lighter one.