I'd agree about the cats. I backpacked the Encampment River from Wyoming into Co. and there was so many ledges and hair and bone piles of mule deer that we stopped very early to camp and started late in the morning. We were pretty nervous at dawn and dusk. On more open trails and no threat of Grizzlies, I'd think you'd be fine with the kids, as long as they stayed close and you followed good food and scent protocol.
I could see going into the Green River Lakes and taking day hikes, even short backpacks pretty easily. Beautiful area, pretty easy access and good fishing for fun with the kids if you like to fish.
http://www.fs.fed.us/wildflowers/regions/intermountain/GreenRiverLakes/