If your daughter is eager to join you hunting, I'd say at least set her up with decent equipment if you want to encourage her.
If I couldn't swing a proper rifle for her, I'd find a second stock for one of the 270 or 30-06 already in the inventory, and cut it down to fit her, before going the handi-rifle route.
Don't send your daughter off to hunt with equipment you wouldn't want to hunt with yourself. You can find a stock to cut down for much less than the $150 you're willing to drop on that single shot. Or you can hunt the pawn shops and get a used M700 $300 to cut down, then add a used fixed Leupy for another $100. It it's ugly, then spend a week refinishing the stock for her. She'll appreciate the fact it's not fugly, and that you cared enough to put some sweat into her gear.
Buying crap for your kids and pretending it's just as good as what you're using is the same as "talking down" to your kids. They'll smell it a mile away and won't respect you, the gear, or the situation. If you're trying to create a great hunting partner, and help your daughter form a lifetime hobby, buy her gear that helps her have success.
