Okay, I ended up writing some code that uses get_field_object() to build me a nested array tree that I can use to store data. The tree I built by hand had several mistakes in it because of how complex/convoluted it is.
Now it’s working.
I have to say that I am EXTREMELY disappointed in how ACF stores data and the extraordinary lengths I have to go through in order to work with complex forms. For non developers, ACF really is great, but for developers working on complex projects, it’s a nightmare.