Weatherby SA-08 Youth/Compact would be my first choice. My grown daughter likes hers better than the Winchester SX-3 Compact Walnut Field she had.
Mossberg Youth SA-20 is good but some of them don't cycle light loads well.
Franchi Affinity Compact is a great shotgun but will have more recoil than a gas gun.
Franchi 720 Compact is great too but hard to find since they've been out of production for a while. They are pricey when you do find them. AL-48 would be too light IMO and some people don't like the barrel movement.
If you want to up your budget then a Browning Silver Micro Midas or Winchester SX-3 Compact Field.