Should I buy an additional battery?
The short answer is in most cases, no. Batteries need to be done properly or not at all. You can get away with spending 10% less on a solar system and it can still work. Batteries on the other hand, are far more complex and have a lot more things that can go wrong.
Furthermore, a 1.2kw battery is not enough to even run many people's tiny split cycle bedroom aircon, which actually requires over double that power to turn itself on and get to temperature.
So, what's the solution? Spend the extra few thousand on some more panels and get the extra production and feed-in tariff. It's far more intelligent from a financial standpoint. However, if you want a talking point for your home with friends, and neighbours, then a small novelty battery will be a popular conversation and make you the life of the party!