If this does open with the Zod fight in MoS, this is a very clever way of setting up the dynamic that's always existed between the pair. By the end of MoS Superman has basically sided with the government and the government believes they can control him, but Batman believes that superman is uncontrollable and goes against the government.
Personally, I think it makes more sense reversed.
In a realistic setting, a Superman outside of government control makes sense, but a Batman operating for decades without some degree of government tolerance is doubtful. Post-Crisis DCU and Timm's DCAU both acknowledged this. Batman's identity was known by Amanda Waller and the Bat Family operated at the tacit approval of Checkmate, ARGUS, Project Cadmus, DEO, Stormwatch, etc. Classically, Batman is basically a policeman or special detective at the beck and call of the Commissioner and the Bat Signal (or Bat Phone). In Begins, most of Batman's gear comes from Department of Defense contract R&D. Most variations of Alfred have him as a former government agent versed in spycraft. Towards the end, "Batman, Inc." was basically a multinational corporation with government liaisons, embassies, and interests, including private police robotic police forces.
Between the two, it makes much more sense for Batman to be a government operative.
Using James Bond as a template, all the logistics of Batman fade away and are quickly intuited by audiences without need for lengthy explanation. Where does he get all those wonderful toys? Government R&D. Where did he get his training? Where secret agents get theirs. How does he keep is identity secret? The same way the government keeps the identities of those in deep cover, in witness protection, or their spies secret. Doesn't he act outside the law and rogue sometimes? Bond has a license to kill and acts rogue
most times. Why is Batman a ninja? BlackOps and SpecOps are basically ninjas.
Also, look at it from a match-up perspective... Superman plus the Government versus Batman? Grossly one-sided against Batman. The only way Batman even manages to land a punch in TDKR is because Gotham descends into anarchy removing the government from the equation... Batman essentially institutes martial law and
is the governor of Gotham. However, Superman versus the Government plus their ultimate super-agent? That's a title card!
Of course, the government doesn't need to be monolithic. I can see Superman being the ally to all those aspects of the government you learn in civics class which reside in the light and based on ideals from the era of Enlightenment... conversely, Batman can be the embodiment and product of the dark underbelly of the government where pragmatism, paranoia, and preparedness trump lofty ideals. Being two sides of governance sets up a situation where their ideals can conflict but where reasonable minds can differ.