The achievement is missable, but the part with moving the switches is not. This is because you have to do this to progress through the story, but you might do it in more than 15 steps.
I skipped every cut scene, and puzzle in the game except for this one, and the one for Master Cartographer. I spammed the click button while searching, and I paused the game. But I played through the entire episode without quitting out and in one sitting.
The steps that I followed:
I moved the topmost switch 7 times.
Then I moved the switch below that one once.
After that, I moved the switch below that one twice.
And finally, I moved the bottommost switch 5 times.
That ended up being 15 steps.
I however, did not get the achievement when I did it. I decided to stop my video like normal, play through the game, and come back to it after. When I completed the episode though, after the cut scene and before the credits, I got the achievement for completing the episode, and this achievement too. I did not even see the notification for this achievement. But when I refreshed my TA achievement tracker, it was there. So I went on Xbox.com to check, and it was there too.
Here is a video showing how I did the steps.