This is probably a silly question but have you been able to load test the motor? I know you can't touch the equipment but I wonder if using a "clamp on" to test current draw is considered acceptable?
As I am sure you know if all three phases are balanced then it would be a pretty good assumption that the connections (internally and externally) are ok.
Is it possible that the soft starter is not adjusted correctly? Maybe adjusting the soft start to ramp up the voltage a little slower will solve some of the problems such as the rattle in the conduit and Hid lights dropping out when the motor starts.
It has to be tough when you can't really lay your hands on the elevator equipment...the temptation to dig in and solve this would be a little nerve racking.