If you can find an older model 3 screw Single Six they seem to be a smaller and lighter gun than the new model SS's. They are a little larger than the Bearcat but that's a good thing. For now use some files or sandpaper and make the lower portion of the grips smaller to let her get a good hold on the gun. Google for "gunfighter grip shape" or some such thing. It's a common mod to thin and put a more rounded curve on the base of the classic SA gun grips. And in this case it'll greatly help her get a better fitting hold.
As she grows you can replace the grips with some nice new ones and you and her can keep on using this great little gun.
Folks with Heritage guns are mostly happy from what I've seen. But they are not as nice to shoot in many cases with grittier triggers with more creep than the better guns. Now a little internal work can turn the "cow's ear" into a "silk purse" if you're up to the job skill, tool and patience wise. Otherwise I'd suggest spending a little more and get a better gun from the beginning.
One budget option to look into is some of the older used guns. Hi Standard made a Double Nine that when used as a 9 shot swing out single action. But I'd tend to avoid them. I had one and even the SA trigger pull is overly heavy. H&R made a wide range of top break and swing out light duty target .22 revolvers as well. I've got a Sportsman top break and just bought a 922 swing out yesterday. Both light guns with decent triggers which would suit a 9 year old nicely.
In terms of muzzle control I'd tend to avoid the top break since things get pretty funky about how to hold it to avoid muzzle sweeps when the gun is broken. However the swing out cylinder of a 922 is a good option for learning to deal with maintaining muzzle control while performing the loading and unloading.
The S&W K frame .22's are SUPERB guns. But heavy and fairly bulky for small hands. A model 34 or 63 on the other hand would be nice. But if you want to avoid double action for now then just skip on these.
Taurus certainly seems to be hit or miss. If you get a good one they are really nice. If you run into troubles they can take repeated trips back for warranty work. So it's a bit of a gamble. Also they are all DA which you may be trying to avoid.
So all in all in order of ideal options I'd suggest the following;
- Used older 3 screw Ruger Single Six. If needed thin down the grips to suit her hands. Replace grips when she grows.
- Ruger Bearcat (small for adult hands)
- S&W 34 or 63 (double action, small for adult hands)
- H&R swing out 9 shot
- Heritage Rough Rider