I initially thought this achievement was like all the other area specific gen repair achievements but it is not. Typically you would need to complete the gen (typically) closest to the basement in a special area of the map.
Isolation room I took to be where the upsidedown gateway is and to my disappointment it was not. Completed this gen twice and escaped and thought the achievement was bugged. I should of read into the the name of the achievement a bit more apparently.
The achievement is actually tied to the one man generator located upstairs on the map. I believe you have to be the one to complete it so if you get scared off and someone else takes your place it will not count for you.
You can escape via the gate or the hatch.
I'd recommend bringing a nice toolbox in and burn the map addon that gives you this map. Maybe balanced landing as there is a nice drop nearby that you can use to make a quick get away if you get cornered.