sounds like both parties are wrong to me. if i was the old lady, i would have sat facing the wall (unless, there was some other reason why she couldn’t take that seat – she has a hearing problem and her rav said that hearing through a hearing aid would not be yotzeh, so she has to sit right next to men’s section to hear properly, and she was too embarrassed to explain to some woman she didn’t even know);)…
but if i was the mother, i would have been mevater and put my skinny daughter next to me or on my lap, and later tell her how proud i was of her for giving up seat to lady who obviously had something wrong with her, if she didn’t move when i told her that my daughter was sitting there.
popa – may not have to give in to bullies, but if one time thing, definitely is a mitzva to be mevater and not make a whole big deal and humiliate child and woman.