This is weird.
There are many scenarios that could fall into the bare outline you describe, ranging from mildly annoying to totally psycho.

What does the grandma DO after she weighs the child? What's the context?
I definitely agree that the approaching should be done with the parents of the child and not the other Nana, unless the parents are at a loss and would appreciate the help.
There is a lot of literature (scientific studies and such) that could be brought to bear on convincing whomever needs to be convinced that the weighing is not a good idea. But again that depends so much on context, further information about how it takes place and what is done with the info about weight, to know how to best approach it.
If the child is only two, one can be fairly confident that no lasting damage has been done, and there will likely not be any particular effects if it's stopped soon.
And hopefully it'll stop soon!
Happy to provide more ammo-type info once more context is provided (if it's known/ if Julia is comfortable providing it).