Since I could not determine where you are (an indication of deer size), nor do you mention the ability of your little shooter, I'll provide some general thoughts.
Although many consider .223 adequate for deer, you find most supporters hail from states with smaller deer (FL, TX, etc) or their ability overcomes the weaknesses in the cartridge. I shot wild hogs with a Mini-14 in .223 in Florida growing up, but shot placement was important.
Your choice of .243 is well considered. A plethora of factory ammo exist in the heavier bullets you'd want to choose. It also performs the varmint role well with lighter bullets designed for the task.
.260 is definitely worth a look! Based on the same .308 casing the .243 came from, it launches a larger bullet and does it well. From articles I've read on the cartridge, more shooters are showing up every year at long range competitions with it. Factory loadings are more limited than the .243, and finding boxes of ammo anywhere outside the larger stores might be a challenge, although online ordering is an option.
Some other cartridges exist you have not considered. .257 Roberts is a mild-recoiling round that is very adequate for deer. The hard part to get over with the venerable "Bob" is limited ammo selection and IIRC, only Ruger is currently chambering a rifle in that caliber. If you use Gunbroker, you won't have a problem finding an available rifle, however. Like the .260, you will have a hard time finding a box in anywhere but the larger stores or online.
7mm-08 is another .308 parent cartridge to consider. It is only .5mm larger diameter than the .260, but gets the job done very fine on deer sized game and is more common than the .260 or "Bob".
The .250-3000 Savage was a nice round. No longer chambered in a factory rifle I am aware of, trying to find ammo can be challenging in person, but the "electronoshpere" can assist. Most of your used rifles will be the excellent Savage 99 lever action.
If you are considering an AR-15 platform, there are a dizzying array of available cartridges to consider. Since you are concerned about recoil, this is the best platform for recoil mitigation. However, the loadings for deer sized game present challenges to find ammo for unless you are willing to buy online. A quick search on this topic will reveal a thread started to document all available chamberings for the AR-15. Just beware: I took this route in 6.8 Remington SPC to meet the needs of a recoil adverse daughter, an acceptable cartridge for out to 100 yards, only to be forced to purchase my deer hunting ammo online. Absolutely zero shops in my local area carry anything other than Hornady with V-Max bullets.
Good luck and I hope this provided some help.