On a "business purposes" license, is picking up the kids from our ex-wife's house considered "driving necessary to maintain livelihood"?

No, picking up children from an ex-spouse's  house is NOT "driving necessary to maintain livelihood," except in the very ususual case where you are employed for the purpose of picking of the children, or it is  part  of  your job--e.g. if actually worked for the ex-wife, and part  of your job (as, say, her assistant) is to pick up and drive the children around. In that case, you migh be able to make the argument that it is "driving necessary to maintain livelihood, but not otherwise.