I agree and then disagree...safety course, get the quals, you by her side.
Then get her, or rent at the range, a .22...you live in a city, more or less, so she is not going to be unsupervised at all. Gone are the days when I marched off with my Crossman .760 in the boondocks of Cali looking for trouble.
.22's are zero impact guns from a shooting standpoint, as you know. They pose no threat to the shooter. In my opinion, "something" goes down range from either a BB gun or .22, or a Crossman, all dangerous and to be controlled. The rules are the same for all guns.
But .22's are, within the boundaries you'll set, a frickin' scream to shoot. Very tactile and satisfying.
I started my kids on .22's, then moved to .410's. Here is a picture of my son when he was 11 with a 12 gauge Browning A-5. Very little recoil and he had worked his way up. Ignore the trigger finger, we fixed that.
Neither of my children have shot anyone...I worry more about my daughter on the back of a horse than shooters.