Depends on the bullet, but they’re both close. For same weight bullets the smaller caliber will have better BC’s. A .243 can be loaded with a 115-120gr bucket with a higher BC than the 120gr 6.5, but the 140hr 6.5’s have a higher BC than the 120gr 6mms. The 7mm 140s have a lower BC than the 6.5mm 140s, but you can go all the way up to a 190gr bullet in 7mm and it has a much higher BC than any 6.5. Factory twist rates in the 7-08 don’t allow the heaviest bullets though. A 7mm 168 has a similar BC to the 140’s from a 6.5.
What it really boils down to is are you interested in handload, and do you want a gun that will push heavier bullets one day when she is ready? The 7-08 will push a 140gr bullet faster than the 6.5CM and at normal ranges BC doesn’t even matter. On top of that it will shoot 168s very well. For low recoil, you can even shoot a 120gr ballistic tip hunting at deer and for practice. For lots of factory ammo in the 120-140gr range, you might lean toward the 6.5CM. The 7-08 is a little more versatile of hunting cartridge though if you don’t mind hand loading because you can find hunting bullets from 120gr up to 190. With a 10 or 9.5 twist you’ll be limited to something in the 168-175gr range at the top. With an 8.5 or 9 you can shoot just about any bullet out there.
I have a .243 and have used on mule deer with success. It’s a nice little rifle. However, Most would recommend using a 105-120gr bullet deer or elk. If you do that, the recoil will be very similar to the 6.5 CM or 7-08 with a 120, but that will be as heavy as you can ever go and barrel life is shorter in the smaller bore, so if she gets pelenty if practice you could loose some accuracy over the years. I would choose the 7-8, but the 6.5 CM would an excellent choice as well. On the secondary market you could probably pick up a .269 Rem for a good price. It’s everything the 6.5CM is but because it isn’t the new kid on the block they’re not as popular.
I agree with the heavy (but not too heavy) bullet per cartridge concept. This is why I shoot 180 in my .308. If I wanted 150 gr, I'd be better off with a 270. Even the 165 gr only gives me 80 fps more velocity and 1/3 inch less drop at 300 yards with 200 yd zero. Not to mention less energy at the muzzle, even less down range due to bc and momentum, and slightly more wind drift.
Not that any of this really matters in real life, but if you gotta choose 1...
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