The issue here is you are trying to match the $20 for one battleship which gives us odd comparisons. For a real comparison look at multiples of each where the dollars match.
a) (5) Destroyer (5 HP, 10 OFP, 10 DFP) vs (2) Battleship (4 HP, 8 OFP, 8 DFP)
Destroyer versus Battleship = 59.7% DD win, 35.2% BB win, 5.1% both die.
b) (3) Destroyer (3 HP, 6 OFP, 6 DFP) vs 2 Cruiser (2 HP, 6 OFP, 6 DFP)
Destroyer versus Cruiser = 66% DD win, 27.3% CA win, 6.7% both die.
So, in a straight up battle the destroyers win hands down. If you need defense the destroyers are great. If your fleet is already bigger than subs are even better on offense than above. Subs and destroyers are better than cruisers and battleships.
I have bought battleships when I only have limited space in the factory slot(s) and needed the biggest defense available due to an enemy fleet threatening me. Such as Japan off FIC or the US off Korea. I may have bought a cruiser once in my lifetime in the same scenario when I only had enough space and money for the cruiser and could not get the battleship or multiple destroyers or a carrier and two fighters.