I'd imagine a shoggoth could be like a huge thing that slowly oozes through the level. So it wouldn't be an adrenaline-filled chase sequence but rather this threat that's not as immediate as a brute chasing you, but makes sure to keep you on your toes. Like water thats slowly rising in a level, but less predictable, y' know? Dunno, it might work.
You could also pull a "War of the Worlds" type of scene, where several tentacles are probing around in a room, searching for you...
Or you could have the shoggoth hang around in the levels pipe or ventilation system, so you have to listen carefully for the noise it makes when it moves around and be very quiet yourself, or it might stick it's tentacles out of a hatch to grab you
It could also serve a more passive role, like a natural level barrier. Let's say the story takes place on a ship - well you'd need a reason why you're not simply escaping over the water in a lifeboat, right?
But yeah, I think it would be pretty hard to properly integrate a shoggoth in a way that makes sense, compared to more "normal" enemies. It would maybe best be used as a more subtle "background threat" I feel...your main problem are the smaller enemies but you also have to be careful not to wake up the huge, slow beast thats sleeping somewhere below you. ^^