I haven't posted for a while as I've been doing research on many of the models listed here, but I've been continue to keep up with the new posts. I'd like to thank everyone again for the continued advice and ideas in this thread. This has been extremely helpful in giving me a baseline of options to consider.
At this point, I've come to the realization that what I'm looking at here is at least a two, if not three stage process. I'm almost settled on a Bearcat for the first family/child handgun, for all of the reasons already discussed. This is not only for my daughter, but for my 7-year old son as well, who will definitely benefit from a slow, careful, and most importantly safe experience with handguns. While I'd love a less expensive option, I'm starting to think that it makes more sense to spend more on something that will last, and even be enjoyable to shoot down the road, than something less expensive but of poorer quality.
If that ends up sticking, it'll be a fair few months to work a Bearcat into the budget, so I've got time. Time to let her continue to practice with the CO2 pistol to build up strength, with occasional dry firing or even live firing of the Mosquito. Not to mention just making certain that firearms/handguns hold her interest at all.
After that I have a few decisions, and that will depend a lot on where the whole family is by then. If I had to choose today, it would probably be a Neos as a second "family gun" rather than something specifically for my daughter. But that may well change, and I want to get a chance to handle (and have her and my wife and son handle) some of the other options first. This is at least 12 months away unless we win the lottery, so there is time...
Finally, I figure I've got at least 2-3 years (both for familiarization as well as budgetary reasons) before it's time to let her pick out a gun for herself. It might come sooner, it might come later or it might come not at all, but when it happens, I really want her to have a good idea what she wants to do. If she's interested in plinking and target shooting, then it's her very own .22 in whatever style she prefers, but if she wants to do more "combat" shooting, then she should have had the opportunity by then to know what she wants in caliber and not just frame.
I will continue to read replies here of course, and whenever I make the final decision and make the first purchase, I'll try and resurrect this thread just to share the results.
Thanks again for the great welcome to the forum!