Lanchester's square law works in the opposite way that you applied it, a square rather than a square root. By that analysis (which doesn't take into account most things in SS like regenerating shields), a 30% increase in firepower is a 69% increase in effectiveness. I don't think this is accurate for SS, but its more accurate than taking the square root.
Actually if I'm understanding the wikipedia article on it correctly, Lanchester's square law should be a square root in this case. The law is that twice as many troops represents 4 times as much effectiveness (or firepower). Since the number of troops you can have is based on your DP, then yes, 30% more firepower represents 14% more DP.
However, I'm not sure if Lanchester's square law can be directly applied in this case, due to Starsector's game mechanics, and due to AAF's burst nature. AAF's damage is front loaded, meaning that during (roughly) the first 6 seconds it does double damage, then during (roughly) the next 11 seconds it does normal damage. (I'm ignoring the whole chargeup/chargedown thing for simplicity here). So the average damage dealt is basically 200% for the first 6 seconds, then gradually tapering down to 135% at the end of a duty cycle. So it only ends up averaging 35% more damage at the exact end of a duty cycle, right before the next burst starts; if the enemy ship dies before that, i.e. in the middle of the cycle (which is virtually always, unless you have really bad luck), then the average damage dealt is actually greater than this.
Also, it does roughly half of the total damage per duty cycle during those 6 seconds, meaning there's basically a 50% chance that the enemy ship dies during those 6 seconds, and a 50% chance that the enemy ship dies during the 11 seconds where AAF is on cooldown. So yeah, on average, when the enemy ship dies will skew heavily toward while AAF is active; it's not evenly distributed throughout the duty cycle.
Thus, in actual combat use, chances are AAF does more than the +35% more damage that the duty cycle implies; the +35% is actually just a minimum bound on AAF's effectiveness. The max would be +100% of course (if all ships die during AAF's burst). So, even if Lanchester's square law applies in this case (and I'm not convinced it does), all you can really say is that it means the ship would be somewhere between sqrt(1.35) ~ +16% and sqrt(2) ~ +41% more in DP than if it didn't have AAF, depending on where in the duty cycle the average ship it faces dies. (And you'd have to take this averaging across multiple duty cycles into account versus harder targets.) That means that if the normal AAF version changes to 22 DP, then the pirate, non-AAF version would be between 15.56 to 18.93 DP, depending on your assumptions of where in the duty cycle the average ship dies, and how much you feel burn drive improves the ship compared with not having a system at all.