Bowls can keep bettas alive, but not in great conditions. Keeping the temperature stable is the primary problem. The second concern is that without regular (ie, daily, or every other day) water cahnges, the tendency is to completely clean the tank once a month, removing all the beneficial bacteria.
I would have to say "If you are not going to allow her to have a tank that can properly provide for the fishes needs, a bowl is not an appropriate substitute for the animals. It would be better to do this right, and will be glad to help with that." And stand firm on that--when the Mom does the bowl and the goldfish, and they die, maybe she will talk to you and try doing it right rather than putting up with a very unhappy child.