The local grocery store also operates a series of gas station/convenience stores. While they carry the same name and branding, they each operate independently and set their own policies. For example, one store discounts for soda refills while another does not. One location has a punch card that requires 12 punches for a free drink while the others only need 10. I think they have as many differences as similarities.
When I go into a franchise, I expect consistency. Part of the reason people go into McDonald’s is that they know they will get the same thing at any of its locations. Other franchisees strive to achieve the same. Organizations with regional or functional chapters are essentially “franchises” bound to follow the same principles.
Don’t confuse your customers. If you want to function with autonomy you should operate as separate organizations under different names. A franchise — or subgroup of a larger organization — requires a commitment to the whole. If you take the name, you take the entire package.