I agree there's no need to worry about food getting digested - that will take time, but it will come. However, I definitely would not give raw carrot, especially those baby carrots - they're basically shaped like perfect little throat-plugs. On the bright side, they're very quick to cook - a few minutes in a covered saucepan with a small amount of water steams them in no time.
Also, as I understand it, one of the main (and only) rules of BLW is to have the baby sitting upright (i.e., in a chair, or on your lap), so I would say feeding tidbits while carrying her on your hip is not such a good idea. They're supposed to be sitting up and having an official mealtime because: (1) that way when they gag, their position is such as to help the food get safely spit out, and (2) you'll be in a position to notice when something is not getting swallowed properly.