Hoosier, assuming you will be staying on or near the AT (there are shelters off the AT, but not many), the GSMNP "dollar" map shows where on the AT horses are allowed and not allowed.
My favorite shelter is Tricorner Knob, but its also a favorite stop for horseback riders who have ridden the Balsam ridge from Catalooche.
I don't know your kids, but I know if I was a kid (kindof still am), I would rather stay in a tent with my family than be forced to sleep with a bunch of strangers in a critter infested shelter.
Remember that peak times for Northbound thru-hikers in the smokies is right around the corner.
Have fun! its good to hear about parents taking their kids past the front country campgrounds.