There's a specific sequence to this that I don't think anyone has explicitly mentioned.
Upon entering the room with the Tinker, in one corner of the room (across from the prison cell) are a set of three computers.
- One computer is larger than the others. You must use Computer Science on this first.
- Then you need to check the other smaller two (right click to look at the terminals). One will be a "capacitive overload" and the other will disable the "wi-fi." Use Computer Science on the wi-fi computer.
The larger computer can be destroyed by shooting it, but this kills Binh. Hacking the other computer also kills Binh. When Binh dies, you'll get a "Log Book Updated" notification. Doing the "wrong thing" makes the fight easier by damaging the robots and/or applying negative status effects to them.
I was also careful not to shoot towards the larger computer, lest I damage it by mistake.
Tinker will also explode when destroyed, so I made sure she(?) was away from both Binh and the larger computer prior to killing her. (I don't know if this mattered, but it seemed worth it anyhow)
I did the hacking before combat initiated. The "wifi" hack caused Tinker to notice me immediately, so set your team up for combat before doing this hack. You can also do the hacking while combat is going, it takes 4 AP per attempt.
Tinker will also pause when you get her down to critical health, although talking to her made her blow up in my face, so on my final run through I just shot her rather than attempt to initiate conversation.