You are right, she is wrong. It's not worth fighting over. If she puts you down as a reference for any of these jobs you can always tell the truth if you're asked if there is anything you want to add to your answers.
But she will need that daytime child care no matter what shift she works because she will need to sleep every day she is working nights during her child care hours. It is fact that people need to sleep. If she has her child home with her during those hours she will not get to sleep.
So her child will be going to child care those hours no matter what schedule she is working.
I think that if she has a good support group that will help her to become employed and earning an good income she will be able to have a better life and more able to provide for her child.
My oldest grandson's father worked shift work and my grandson went to school every morning when dad picked him up and dad's mom's house where he spent the night. Then dad took him to school, then he went home and went to bed for his "night time". When son got out of school dad was there to pick him up, they had their evening together and dad dropped him off at his mom's at bedtime. He even did the bedtime routine with him and got him asleep. If there was time he'd go take a nap before going to work.
This plan worked very well. When my grandson was out of school he knew dad was there but he was sleeping. If he had an emergency he could wake dad up but dad's mom was a few houses away and he'd call her then go over there most every day so dad could sleep.
If your friend has anyone that will help her during the nights with her child then she actually has a good idea about child care during the day though.