In my view the only way to continue BSG with any connection to either the original or Moore's version is to jump fully into the "cycle of time" concept introduced in NuBSG. The finale insinuated that this 'experiment' is ongoing in the galaxy orchestrated by whatever entities mortals believe to be gods. Technically an infinite number of Galactica stories could play out, with only subtle connections to previous iterations of the cycle of humanity vs synthetic life.
If there was to be any ongoing plot between them it would have to be meta. Such as hints to the greater purpose that the people of each cycle discover before their inevitable end. I don't think it works so well to try various spin-offs and side stories set within any one cycle. The core story of Battlestar Galactica is kind of limited in flexibility - it's about the last surviving ship of a civilization leading survivors to a promised land. Any side-stories have to service that premise and can't really go off on their own.
If you were going to change that, it'd have to be right from the start. I.e. one of the cycles of time would have to play out entirely differently and not be about a genocide, etc.