Want to send a note telling mom to get a bike and they both go for a real ride, there are country roads around here.
Lee, I know you feel bad for the girl and your intentions are good and come from the heart, but a note like that would not be very well received. Since the mother is also overweight, she may take personal offense at your suggestion that she ride a bike too.
Also, unless it's someone who is used to riding on country roads, it's not reasonable to think every woman is willing or able to do that. If they are both out of shape, they probably wouldn't even make it back home, you have to work up to something like that, make sure you ride well enough to be safe around cars, etc. And you may have to be wary of your safety if the road leads to a remote area.
If the mother is too quarrelsome to speak to anyway, then you'd just be poking at a bee hive.
There were some suggestions to speak to the girl too, I don't advise that either. Unless you were already friendly with the girl and had short chats about other things in the past. The girl would be uncomfortable answering questions, and afraid of making her mom mad. The mother would probably notice that through the window, and give the kid the 3rd degree about what she said to you and what you asked her.
Sometimes we see something that bothers us, then dwell on it and let our imaginations make it into something bigger than it really is. That could be happening too, I'd just ignore the girl for awhile, if that's the only exercise she gets all day, it's okay if she's a little bit tired, IMO.
I would contact Child Protective Services. That is child abuse.
I am complete against involving Child Protective Service for a mother making her overweight child ride a bike. Those agencies have serious abuse cases to investigate and deal with.
Not fair to the mother to put her on an abuse list for just assumptions that the bike riding is just part of real abuse behind the scenes. Neighbors who have falsely reported parents, regardless of their reasons for doing so, often bring unwarranted grief to the innocent parents.
I think it's invasive, and would never report anyone for something as terrible as abusing their child, unless they were beating, starving or chaining the kid to the bed post and not allowing them to see the light of day outside. Bike riding outdoors is far from abusive.