It sounds perfectly survivable to me. You're well kitted, and perfectly well equipped to win that fight with high likelihood and minimal damage. I've been in almost the exact same situation.
Step 1: Ion set to autofire at shields, keeping them suppressed and allowing the attack drone to deal damage. With an automated reloaded, this 20-30 seconds, assuming a fairly normal evasion rate. Since this is a mantis ship, with a teleporter, disable your o2 and vent empty chambers. You don't NEED to do this, but it certainly helps kill them faster.
Step 2: Mantis teleport in. Pull all power from o2 if you haven't already, open all doors creating as many paths as possible to the o2 chamber from the vacuum. This may involve relocating units - so be it, you have mantis boarders, you have to deal with them first. The boarders will stop attacking the o2 as soon as the room goes vacuum, and Mantis can NOT deal enough damage to destroy a system before that happens (although they can get amazingly close and make it susceptible to any attacks that get through, missiles would have destroyed it anyway and it will be fully restored the moment they leave the room unless that happens.
Step 3: As soon as their room goes vacuum, immediately close all the doors. They will start trying to fight their way out of the room - if they are trying to fight their way into another room that isn't vacuumed yet, vacuum it. The exception here is the medbay : a level 2 medbay should allow guaranteed success versus even full life mantis with a human and engi, and a level 1 with 2 engi should even be able to take out the now injured mantis without any problems. If you need to, retreat your other unit to the pilots seat - he can help keep that functional and continue giving a chance to evade missiles, though you will surely eat a few.
Cautionary Step: If you see a missile about to hit your door controls, VENT EVERYTHING BUT PILOT SEAT AND MEDBAY. Especially with the nanorepair, just keeping those two rooms functional and with a bit of o2 will be more than enough to fix whatever is broken after the fight.
Step 4: Close doors, begin re-oxygenating the ship unless they have more boarders to send over. If that's the case, close all doors and leave Oxygen off. There's a high chance of them hitting a vacuum room or, if not, being able to vent their room much quicker.
Cautionary Step: Do NOT attempt to make repairs to any system except drones, weapons, pilot seat and engine, and only if the rooms have oxygen and your units have full health. If it looks like you can win, prioritize drones and weapons and ignore engine. If it looks like things have gone south and you need to pull out, prioritize engine and ignore drones and weapons.
Do all that and you should have been fine. I mean, lucky missile shots (like 2 strong missiles to the medbay your engi are in) can still screw you over, but it's vanishingly unlikely.