If only Ruger would make a version of the BearCat
with adjustable sights.
Since they don't, any of the S&W J-frame series is your next best bet. I would avoid the latest snub-barreled models, particularly w/fixed sights. Besides the fact you have no way to tailor it to your ammo, their triggers are notoriously hard. A used steel M34 in blue, or a used M63 stainless should all be found for around that $300 mark. For a bit more, the M43 in blued steel with an Airweight alloy frame is also a great choice. The size and weight of the excellent K-frame guns may be too much for your kids. You'll have to check for yourself.
In autoloaders, there's really only two choices: Ruger and Browning. Ruger's steel frame MKIIs have a smaller grip, but their mags are a royal pain to load. Guess who your kid's will turn to for that chore?
The plastic frame 22/45 is a lot lighter, less fatiguing and has much more thumb-friendly magazines(
No, they don't interchange). IF your kids can get a good firing grip on it, that would be my suggestion in Ruger.
Browings have alloy frames for lighter weight, and easy to load mags. They've got a model particularly suited for kids called the Challenge. The grip-frame is reduced in size to be even smaller around than the MK Rugers. The tapered barrel is at 4", IIRC. An added plus with BuckMarks is that you can see the chamber for safety no matter which side the gun is laid down on.