Nothing wrong with either choice in the right package that balances well. The primary variable(s) at this point is your daughter's changing stature and the game animals that the two of you will pursue. A rifle that is too heavy will preclude wandering too far afield and the .243 would be considered by many to be too light for the likes of a heavy-boned bull elk....

If you roll your own ammo, there may be other options with light recoil and proven terminal performance. I have tipped over elk with a magnum and have also harvested mulies with a .270. I'm liking the weight/balance/ballistics of a quarter-bore -06 with a 100 gr. TTSX over an ample dose of R22 and a Federal 210M go button for deer & elk. It carries easy, and the X's at 3,400+ fps will seal the deal. For your initial choices, go 130 gr. X's in the .277 or 80/85 gr. (TTSX/TSX) in the .243. Federal loads 'em too....

For glass and light rings, you'd be hard pressed to beat Leupold on Talley Lightweights or Leupold 2-pieces. From lightest to heaviest-- 6x42, 2.5-8x, 3.5-10x with a standard duplex reticle. The fixed six may be a bit much for westside bush whacking. Lacking an elevation turret (M1) on the above, sighting in at around 2.5" high at 100 yards will put you right on at 250 and around 3 inches low at 300 yards. POA = POI for lung shots out to that distance. Federal's website has ballistics tables and a calculator for their ammo. JBM's trajectory/drift calculator can be downloaded for refined analysis of hand loads....

http://www.jbmballistics.com/

Have fun hunting for a new shootin' iron!