I’ve now beaten FTL on Hard with the Lanius B, the Kestrel B, and the Engi C, and I can present something in the way of general strategy for beating the flagship on Hard.
On Easy and Normal, the four weapons rooms are isolated, so the best approach is to kill the crew in those rooms and disable the weapons, then attack the shields and reduce the hull to 0. Alternately, if you have a strong boarding contingent and you aim to kill the crew, you should leave the guy in the laser weapon room (second from the left) alive so that the ship AI doesn’t take over and make the ship start repairing itself.
On Hard, the two middle weapons rooms (lasers and missiles) are connected to the rest of the ship, making it impossible to take the kill the guys/disable the weapons approach. These are also the most hazardous of the four weapons, particularly the missiles. How annoying.
So, I think there are two general strategies to use, the boarding approach and the non-boarding approach.
(1) Boarding approach: Kill the whole crew as fast as you can, except for the guys in the two outermost weapons rooms. This way, you can destroy the two dangerous weapons rooms without fear of them getting repaired (at least until Phase 3), and you can use your boarding crew to destroy helm control and shields. For this approach, you will want to kill the crew on Phase 1. Unfortunately, the fact that the flagship has Cloak can make it impossible to pull your guys out before they get killed, and Door Control means that you can’t have your guys run around different rooms to stall for time until the cloak wears off. Thus, it is preferable to have a clone bay. (Playing on Easy and Normal, I usually wait until Phase 2 to kill the crew, since I don’t have to worry about cloak.)
(2) Non-boarding approach: You’re not going to be able to keep the two most dangerous weapons rooms disabled for very long, so you have to kill the flagship as fast as you possibly can. For this, you will need hacking. Level 3 hacking can take down 4 shield bars, and you can use this opportunity to hit the shield room hard-- your weapons will probably recharge before hacking does, so you want your next volley to have a prayer of damaging the flagship without using hacking. For Phase 2, you’ll want your first volley to hit Drone Control to disable the Defense 1 drone that will try to shoot down your hacking drones. Once the Defense 1 drone is disabled, you can send your hacking drone to the shield room, and repeat the process above.
For both strategies, you’ll need to be able to tank, so it’s good to have cloaking and/or a couple of defense drones.