I know they take demons very seriously, so who in the congregation do they go to when they need to bring out the big guns?
JWs don't believe in exorcism, so I'm not sure what you mean about "bring out the big guns".
In my experience, if someone believes they're being plagued with demons or they have problems with hauntings, things moving by themselves or hearing voices or footsteps, that person (please bear in mind that I am talking exclusively about my experiences) also hear voices, has invisible friends, have severe OCD and depression, and/or has been through severely traumatic experiences (suffering PTSD).
As much as I want to take the issue of demons and hauntings seriously, I have no reason to do so based on my own experiences and the people I have heard talking about it.