Adding a battery backup power supply
There are a couple of points to consider with this question. First, you will need to make sure the PSU is correctly adjusted to the float charge voltage of the 12V battery. SLA should be safe to float charge, but over voltage will damage it over time.

One alternative solution (since powering the heaters will limit your run time) is to detect power failure, cut the heaters immediately, and save state to EEPROM in the firmware. This is the approach taken in the latest Prusa printers, and is a bit more complex, but might turn out to be more reliable.

If the problem you are solving really is brown-outs, then a simpler solution might be to isolate the high current and control sides of the circuit. The MCU will be run from a regulated 5V or 3V3, so a large capacitor (with diode isolation) on that regulator's input would do the trick.