What did you get for your kids to sleep on (think 1 y/o)?
I think you could just grab a pillow if she sleeps on her back. If you forget to grab a pillow, she could just use you as a bed. If her home is destroyed, this might be the best bet because she will want something familar and safe. I live in a tornado play ground as well. Luckily, we are rural enough that there are rarely any deaths or injuries from a tornado.
Entertainment (crucial for that short attention span)? Books perhaps, she loves them and they are flat/lightweight.
I think books are the right track for this. My little girls loves books as well, to the point that she rarely plays with any of her toys. I know that when the time comes to bug out this might be feasible, but I would try and grab a favorite toy of hers, again something familar.
Something handy to carry the child in/with that can pack easily? She's just about outgrown the baby bjorn.
Well, baby bjorn or something simlar is about all I can think of. If she hasn't completely outgrown it, then it would still work for a day or two.
Maybe one of those baby backpacks, which you could also use to pack stuff in. Harder to pack, but she might be more comfortable.
As far as this being enough or too much, that is up to you. We have a baby bag for each kiddo. Each bag pretty much stays packed enough to last for a couple of days.
The only other thing I can think of is a blanket. We have a bunch of 4` x 4` baby blankets that the mother-in-law made. The are two layers of fabric. Fold really small, and are really handy.
Just my $.02