12 Volt DC Power Supply set up
You can, and it will probably work.

BUT

A 12V battery does not provide 12V. A 12V lead acid battery (which is what I assume you mean by a 12V car battery) varies its output voltage from almost 15V down to around 11V during its discharge (these numbers vary depending on the type of lead acid battery you have). This means that your power supply will provide pretty much none of the current when the battery is fully charged, and pretty much all of the current when the battery is nearly empty.

The better way to deal with the surge current demand would be to have a large enough capacitor on the 12V rail, next to the amplifier which is causing the current drain. The specification of this capacitor will depend on the load, and on the capabilities of the switch mode power supply.