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The problem is she's small. She still wears pants sized for 8 year olds, and I think she weights about 52lbs.
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My daughter is about the same size. Her big issue is getting comfortable with the LOP if not using a red dot. Here are some options you can look into. Understand what her LOP should be before looking, that is more important then anything
Henry mini bolt youth rifle - it has a short LOP at 11.5" - downside is that cycling the bolt doesn't cock the action, my daughter doesn't have the strength to cock manually
Cricket youth rifle - 12" LOP - same cocking as the Henry
H&R Handi rifle youth - 14.5" LOP
Savage Rascal - 11.25 LOP - same cocking as Henry and Cricket
Marlin XT-22Y (youth) - 12" LOP
Mossberg Blaze Youth Bantam - 12" LOP
Thompson Center hot shot youth single shot - 11.5" LOP
Savage Mark II GY - ? LOP
CZ 452 scount - 12" LOP
What I did and my suggestion:
Find a cheap vintage .22 bolt action. There are a ton of them out there, often for $100 or less. Cut down the stock to her LOP and go from there. You could also do this with a modern lightweight Ruger, Savage or Marlin. If its a common model its easy to find replacement stocks when she outgrows the LOP on the stock you cut.
Not my rifle but it is my daughter. She is using a donated cut down Remington 513T at a local small bore class she goes to. Its what gave me the idea to just find a dirt cheap (note the 513T isnt dirt cheap) used .22lr and cut the LOP to fit her.
The last advice I would give is to pony up for an M&P 15-22. You can find them used for around your price. The LOP is 10" when closed and its light. She can also grow into the LOP due to the adjustable stock.
Here is her M&P 15-22. You can see the 10" LOP fits her almost perfectly.