GB-
The weight and action style of the first rifle my daughter uses are actually almost non-issues. To start with, she will be shooting from a rest, one shot at a time, while I operate the action. Her job will be using the sights and trigger. She will not be carrying the rifle, or hunting with it, or even shooting it off-hand for quite some time, although she's very good at shooting her lever-action BB gun off-hand. Length of pull and comfort at the bench are the main factors.
The Chipmunk, Crickett, and Henry Mini Bolt are all about 2" too long in the length of pull for her to shoot comfortably (we've checked). Taking a saw to the synthetic Henry is out, so that one is not a current option. Even the tiny Crickett/Chipmunk rifles would have to face the saw. Henry does offer a lever action youth model that's lighter than the Browning.
One upside to buying a lever gun would be that the gun can grow with a child as the child's shooting ability grows. The chopped butt stock can be removed and replaced with a new, full sized piece of wood. It won't be left in a closet after 10 years or so, unlike a single shot mini rifle.
Many of us learned to shoot on less than perfectly ideal firearms, but I don't think that a lever action 22lr with a bobbed butt is too far out of line. It will be far easier for her to shoot than my first rifle was for me (full size model 88 in 308 with a hard plastic butt plate).
She has shown that she's mature enough to handle it by handling her BB gun very well, and very safely. She has also shot a (single-loaded) Ruger Mark II very well. She has also shown some aptitude for archery.
In the end, I may very well give her a few rifles to choose from, and let her pick her own favorite from a selection of small 22 rifles. It'll be chopped to fit her, and we'll go have some fun at the range.