What can be done (except shutting down GT) in case of mist eliminator motor failure to avoid oil fumes coming out of bearings?
A pneumatic venture air mover can sometimes be used to establish a partial vacuum - perhaps 1/2 to 1 inch water. The air mover must be properly grounded in a very secure way to prevent static discharge from creating an explosive spark.
I would need to see your system to be sure, but you maybe could bypass the eliminator and temporarily vent to atmosphere. get the vent as high as you can, it should help the leakages by venting the LO tank pressure a bit. You could also arrange the vent so you can remove the eliminator and fix the motor.