I would suggest waiting a while untill your daughter is ready. If she is scared of the dogs at QAWS, no matter how big or small then I think now is not the right time. I adopted a 9 month old from QAWS the day after the fire and we love him to bits.

The temp accomodation he was in is doha vets, they have been fab with me and my new dog and I can tell you they have small dogs and puppies, or they did 2 days ago. If you are dead set on getting a dog then I would suggest going down there and letting your daughter see them and see how she feels. Good luck.