Three crewmen is really, really harsh. As always, there's a small chance you might find a shop on sector 8, if it sells crewmen, buy some. The decision not to sell the Med-Bot Dispersal throughout the entire game turned out to have some benefit, because it will allow your few crewment to avoid medbay visits even if they take punishment from putting out fires and stray shots and occasional lack of oxygen throughout.
You have a lot of important secondary systems stuck on level 1. The upgrades to level 2 can be absolute lifesavers and are very much inside your 89 scrap range - but have to be balanced against the fact that you're also not maxed out in power. You should be able to pick up more scrap on sector 8 by visiting unvisited beacons or selling items to shops; do it. (Sell the hull beam; the Burst Laser II does a more than adequate job wasting the enemy ship once the shields are down, charges faster, and you can never have enough concentrated targeted system damage.) Level 2 in Oxygen, Piloting, the Medbay or Cloaking will make those systems not just sturdier, if you do find yourself in need of oxygen or the medbay before the battle is over, you need them juiced up. It's a tough call. I'd get at least oxygen 2 and invest the rest of the 89 scrap in power bars.
Defensively, you are really vulnerable to boarding. Relatively speaking, the first phase of the boss fight is by far the least dangerous to you, so you should use it to lay the groundwork for an easier time later - try to kill as many crewmen as possible on the main body of the ship and the missiles room. Bring missiles, Ion Blast Mk II and the burst laser. Missile priority should probably be missiles room, then piloting or cloaking or maybe shields, as your ion blast II stays locked on shields, and once the shields are down you just aim the burst laser at whichever system the crewmen are repairing at the time. Use your cloaking device to dodge the first missile barrage. Later, it's probably most useful against the enemy ion barrages.
Second phase, swap out the missiles for ion bombs, and keep the defense drone powered up at all times. Really pay close attention, since it's easy for it to get deactivated by a stray drone. Ion bombs, incidentally, go into the missiles room, so the defense drone can concentrate on keeping the boarding drones away, and your cloaking can stay reserved for the power surges (remember to only cloak shortly after the power surge drones actually appear). Meanwhile, your Ion Blast II should be able to switftly turn off shields and your burst laser should kill piloting first, then perhaps missiles so you can stop using bombs (might need to save up some missiles - but do remember before the flagship to hit up repair bases for extra ones) then the drone control to switch off whatever boarding drone might have made it onto your ship, then whatever else you think is in need of some tender loving carnage. Use cloaking, of course, to dodge power surges. Things might get nasty in this phase - but Ion Blast II + Burst Laser II should kill the flagship fast enough.
In the third phase, if you didn't kill enough crewmen, you might have to resort to venting almost the entire ship (your pilot can't leave; if he gets into a fight, send one of your Engis to assist and pray to Celestia that the med-bot dispersal makes a difference; your remaining Engi will also need Medbay 2 to survive in a 1v3, so make this a priority with your flagship 2 scrap) and retreating to your medbay. This is where having Oxygen 2 might make the crucial difference, since after the attackers suffocate you want the damn thing full of air again so your Engi can repair crucial systems. In this phase, I would actually abandon subtlety altogether. Pump the shields full of Ion and Hermes missiles to take them down as fast as possible, knock out the piloting, and just missile+laser the hull down. Ion knocks out Zoltan shields very quickly, so you should be fine winning the race.