I would be a fan of a hybrid Cutowrm-New Tuscany approach militarily wise for the core and a deep strike approach on the verge.
Identify some of the key nodal systems for Battle Fleet. The ones with the massive construction yards and 50-100+ wallers in orbit (even if some are being worked on). Show up with overwhelming force, and demand the immediate surrender and self-destruction of every SLN vessel in system. If they refuse use your drones and intel to identify the flagship and punch it, and only it, out in one missile launch. Repeat demand for surrender and self-destruction. At the very most I see the GA having to destroy 3 different 'flagships' from 50m out before the 4th in command surrenders. Once the SLN ships are destroyed you leave the system.
Pros: You've just taken out a chunk of the SLN Battle Fleet and blasted their outright incompetence in front of a major system. A major system that now knows that nothing they can do in the next couple of years can protect them. They will panic, and likely either demand peace or to give the Mandarins everything (including a league income tax) that they want. Either way, it will sow discord among the most powerful members of the SL.
For the verge, adopt a policy of striking at the HQ facilities of the OFS. Do not liberate the planet, do not take out sector governors or marines or anything else. Just show up and remove, by force if necessary, their ability to project power to any other system. As more systems start to rebel that will, at the very least, limit what OFS can do to 'pacify' them. This gives independent systems a fighting chance without getting bogged down in policing hundreds of systems (which the GA does not have the resources to do).
Additionally, if multi-system groups (such as the Maya Sector) approach and ask for assistance grant it. There, you can feel a lot better that they will control all internal politics and prevent civil wars and bloodbaths from occurring. All the GA may have to do is send out a squad of Sag-Cs, a division of CLACs, some Rolands and a couple missile freighters that can role pods if necessary (see Spindle, battle of). Especially for a system like that Maya Sector that is will on its way to self-reliance all you may to do is provide that extra force for a short period of time until their own Wallers finish construction. For any other sector polity (say, 3-5 systems that were also close and neighborly) you may need to help them out a bit longer - and maybe mention to them that Erewhon might be interested in building them some ships, or (once Manticore and Grayson yards are rebuilt) offer to build them a slimmed down (e.g., Buttercup era tech) squad of BC(P)s, some Destroyers, and some LACs to let them deal with any pirates or smaller forces that come calling.
