stagger the fire so they both don't fire at the exact same time and you should be fine
I found that manually firing all weapons at around the same time was more effective. For example, I had Halberd Beam+Flak Mk1+Hull Smashing Lasers Mk1+Ion Bomb
Ion Bomb to lower any shields greater than 3, then flak immediately after to bring shields down more, then hull smashing lasers while the shields are lowered, then halberd beam while the shields are briefly lowered. Note I had to pause constantly to get timings right. If I had weapons shooting automatically I doubt I'd have been able to get past the flagship's four shields.
Anyway, beat the flagship for the first time, unlocked a new ship, FINALLY. Only took 40 something tries. Had some difficulty during the flagship's last phase with the boarders. I only had 4 crew, 3 of which were zoltans. I pretty much had to vent all oxygen except for the system rooms. It was real close, there were crew and boarders fighting in the system rooms with fire and oxygen venting out towards the end. Last volley on the rebel ship killed it, but only 2 crew remained, my slug pilot and one zoltan from the original three, the weapons specialist, with 1 health left.
RIP in peace you other two magnificent dudes, your sacrifice will be remembered. I don't even remember your names now, but still.