Ballistically, you're spot on that a 28 bore is way more effective than a .410. Unfortunately, 28 bores are the scarcest, and most expensive gauge so you don't get a lot of gun for the money.
Can I give you an alternative if you have an $800-$1500 budget? Get her a side by side Spanish 20 gauge. It will suffer loss of value (a bargain) if its short already. You can add a 1" pad instead of a 5/8" to get it to grow with her into teen years. You'll spend more on ammo in the short run, finding her 2.5" english 7/8 ounce shells, but it will be a mild recoiling gun. In a few years, you can just switch to off the shelf 2.75" 20 gauge shells.
That's what I did for one of my sons. We got a lovely sidelock ejector 20 bore Spanish gun for about a thousand bucks. It will grow with him and will hold its value.
I would also caution you as a parent trying to introduce your kids to shooting sports. I detest unsafe shooters. I don't particularly like children unless they have very good manners and safe practices in the field. Pumps and autoloaders are rude tools. I don't enjoy watching kids struggle to load them on the lines. (or adults). I don't like getting hit in the face with hot hulls. I don't like getting them pointed at me which happens all the time at clubs because people can't keep their muzzle 100% in a safe direction.
Get a gun that breaks open. It's civilized, safe, can feed and fire diminished loads, and can shoot short shells for less recoil as well. Oh, and again, they don't lose money if bought used.