Measure the units standby current to determine what the consumption is when connected and in 'memory save' status. If the current level is 'acceptable' (determined by the actual current and the length of time you want to save the memory settings for - to get an idea of how large any battery might have to be) you can go from there.
Given the manufacturer didn't fit one by default (they rely on the car battery doing this for you) it may be that the current consumption means a battery that's too large to fit in the existing case of the radio. No reason why you couldn't 'parallel' an external battery though.
There's a pin on the main power socket that is dedicated to retaining the memory - you can measure the current draw there.
Schematic here: