A 12v car battery is not 12v. It is near or slightly over 13v. Put a meter on one. Put a meter on your car's battery while running and not running. It is very high right after starting the engine because the charging system sees the reduced voltage from the high starting current. I like using car batteries as a source because they have ZERO noise in the power. But you have to keep them charged. If you don't mind recharging the battery after every show, it is great. If one battery isn't enough, get two. I bought a car battery that was a replacement for my truck. Off-season spare. I used it on a parade float so that I didn't need a noisy gasoline powered generator.
If you want to run a battery powered show while also connected to a battery charger, the biggest concern from an AC powered battery charger is noise. When charging, a battery is unaffected by any AC noise on the cables, so most chargers make little attempt to reduce it. That noise (sometimes called "ripple") MAY affect your controllers. A Ferrite bead on the power line to each SSC will reduce the noise some, but they aren't terribly effective for 50 or 60hz (low frequency) noise.
When you buy and use a regulated 12vdc power supply, it will be 12vdc, and usually be low noise. Some can be adjusted higher and could match a car battery's voltage. You probably would not see a brightness difference in your LEDs because at 12V or 13V, the LEDs are in a flat part of their intensity curve (i.e., a 10% voltage increase would probably only get a 1% brightness increase).