I think you've finally hit on the real issue here....she's your daughter...but 7 is awful young ....and most all kids that are 7 don't have the upper body strength to really hold, swing or fire a shotgun of any size.
Some kids are an exception.../ some kids grow faster....
The fact that she wants to do it ...is a good thing / but coming up with another game -- with a .22 at that age is probably a better idea / something static ---like putting clay targets on a stick and let her shoot at them or something.
I have a bunch of young grandkids...from 3 to 21 .../ but when a lot of them were going thru this 7 - 12 age group ...what I did was use a 7lb gun I had ..and said ok, if you can hold it properly, shoulder it properly - and fire 2 shells at a fixed target without knocking yourself off balance, etc...then you can come out to the clays range. It was built around safety ...and they all understood the reasoning ...and they knew what it took to "come of age" to go out ...
Some of them lost interest as they got older...some didn't / some tried it a couple of times / none of them really got into it very much.
My personal choice for a first gun for them to shoot was a 28ga ...around 7 lbs...like a Browning BPS Hunter pump gun ...
http://www.browning.com/products/catalog/firearms/detail.asp?fid=011B&cid=012&tid=211&bg=x
3/4 oz of shot, 1100 fps...the gun retails for around $ 625 or so these days ...but its a good investment..good quail gun, etc... I like ported shotguns / but it doesn't reduce recoil much - if any ---its primary issue, often debated on here, is I say it reduces muzzle jump between 1st and 2nd shots.
I think a single shot shotgun is a waste of money ...the BPS Hunter would be a better investment. You didn't bring it up ...but a .410 is a really bad idea...balistically it shoots a smaller pattern, for a new shooter it can get really frustrating...heck, for me, the .410 is frustrating...
She's your daughter...but for my grandkids...they only got to go when they could show they had the strength to earn it.
My own history - I was big game hunting and bird hunting when I was about 8 or so ...and didn't kill my first Elk until I was about 10 yrs old ...but I remember those seasons of being able to go along, help track, keep quiet...before I was able to carry a gun in the field and really hunt. Those days were special too...and I was a big kid, skinny, but around 6' tall at about 8 - 9 yrs old. I got a nice used 16ga, bolt action shotgun, I think when I was 8 ...and still have it ....and I killed a lot of grouse with it as a kid. Its a really light gun ...around 5 lbs...and because its so light / the recoil is "Significant" ....I made it work / but I'm smarter now ...and wouldn't do that to an 8 yr old today ...even one who wanted it badly. Its ok to have them wait ...and earn it. Good luck...however you decide.