Neither Superman nor Batman would be pro-registration. Batman doesn't trust the government (or many people at all) enough to be down with that. Clark knows why secret IDs are important and ultimately isn't going to tell we'll meaning supers that they need to make big personal sacrifices like signing up to the register and losing their secret ID in order to be good heroes. Clark knows that heroism isn't doing what you're told or obeying the law, it's doing what's right and putting things right when there's wrong in front of you.
If Civil War were to be a DC thing then it's make more sense to have all military and law enforcement aligned heroes on one side, superman and batman and likely wonder woman on the other. Hal, Barry, John Stewart, Captain Atom all Pro reg.
I always thought that DCs version of civil war wouldn't be about registration at all - it'd be about the death penalty and whether or not superheroes should kill. Let's say the Joker does something so horrifying that the UN basically says "dead or alive, you have our permission". Clark and Bruce are both, as Kingdom Come got right, two people who don't want to see others die. They'd immediately lead the anti-execution side. All your military and cop guys, arguably Wonder Woman, on the execute side.
Marvel characters have always had that bit of grounding to them, more aligned to the real world, so it makes sense that their civil war was on a matter of practicality and procedure, whereas the DC equivalent would basically be a philosophical battle about the use of force, morality and the responsibilities of the powerful.