Well, let's get to it.
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Phase 1; Cloaking 3, Hacking 3, 15% dodge and full weaponry (3x1 dmg Ion, 3x1 dmg Laser, 3x1 dmg Missile, 2 dmg Beam), no overcharge.
For our current loadout, Phase 1 of the flagship is probably the easier phase to fight. The flagship's hack is on our Drone control, which is something we weren't planning on using this fight anyway. The fight goes; Cloaking 1 to avoid flagship missiles, Hack flagship Shields to completely take down the 4 layers, Artemis flagship Piloting and expend the rest of our firepower on the flagship Shields/Missiles.
Phase 1 ends with expenditure of 2 missiles, 1 drone and no hull damage, of which we were given exactly in the fight rewards. (+2 missiles +1 drone +20 scraps)
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Phase 2; Drone Control, 25% dodge? and 3/4 weaponry (3x1 dmg Laser, 3x1 dmg Missile, 2 dmg Beam), mass drone overcharge.
Compared to Phase 1, Phase 2 loses its Cloak, Hacking, Doors and Ion weapon, but it gains Drone control (Terminator Drone, Defense Drone I, Combat Drone I, Laser Drone I), a mass drone overcharge, as well as additional dodge.
An Artemis is sent slightly before the Hacking drone to ensure the Hack doesn't get shot down by the flagship Defense drone.
Cloak 2 is used to dodge the first missile strike + mass drone overcharge, but I guess the cloak was late enough for a missile to hit our Drone control. Our first barrage barely dents the enemy ship.
Another cloak vs missile strike + overcharge, and this time our firepower makes their mark, as the enemy's Piloting and Shields are completely offline. Upon taking another hull damage from the Terminator drone on our Sensors, Phase 2 is over.
3 missiles, 2 drones expended and 2 hull damage taken. (+3 fuel +1 drone +23 scraps)
A final Piloting upgrade is made, our final battle approaches. (-50 scraps +Piloting 3)