I've seen a number of new pre-teen shooters started off with many different guns. A couple of the best IMHO are the Remington 1100 Skeet guns in 20 or 28 ga.
Although the 410 has very little perceived recoil with target loads, it is comparatively hard to hit with ... discouraging a young shooter in a hurry. Heaven knows,
there's enough adults discouraged with the little "idiot stick".
The 20 ga. will most likely be the easiest to find ... and 20ga. "promo" loads are usually just as inexpensive as 12's. There are enough Remington parts around
that you should be able to find a cheap stock or two to cut down as required, as the youngster grows.
Recoil is pretty much a product of gun & shot charge weight for a given velocity. If both the 12 and 20 ga. guns weighed 7 lbs., the standard 1200 fps target load
of 1-1/8 oz. in the 12 ga. should produce more felt recoil than the same 20 ga. 1200 fps. velocity load of the 7/8 oz. 20 ga. The elt recoil from an auto-loader is generally
considerably less than a single, pump or double as some of the energy of the fired shell is bled off ( either gas or recoil energy ) to facilitate the operation of the action.
I started both my son & daugter off at 12 with a 12 ga. Win. SX-1 (semi), but they were both "big kid"s for their age ( adult daughter is now 6'-1", son 6'-7" )
They both said recoil was not a problem, and no bruises showed. Although the young are keen to get shooting, many don't have the upper arm strength to
do so, i.e., mount, support & swing the gun comfortably. If it's a struggle for them, perhaps wait it out until they're a lttle older. Let them practice "dry mounting" and
"dry shooting" in a supervised safe environment until sufficient strength and ability to handle the gun is demonstrated. If they can get through a "simulated round" of
25 shots over say 6 - 10 minutes without showing fatigue, it may be ready to start. Length of pull and comb height will be just as important to a young shooter as
an experienced one, and as often as not, the young shooter because of their stature will do better with neutral pitch. Good coaching on gun mounting, stance, swing and
forward allowance methods will be a huge help, as will having the gun fitted to the young shooter properly.
And I'd definitely have them start on stationary clay targets well before introducing gentle straight-away risers or anything more difficult.
Good luck in getting her off on the right foot !