If you wish to limit the inrush current to avoid tripping the main or feeder breakers, then yes turning on the displays in groups or batches would be preferable. A delay of a second or two would be fine.
If however the intention is to avoid or reduce maximum demand charges, then nothing is gained by switching on in groups or sections.
The maximum demand is averaged over 30 minutes and therefore takes no account of brief inrush currents.
If last year you were close to the limit, then that suggests that the sustained load was approaching the service capacity, nothing to do with inrush currents.
Maybe not so much the service capacity but rather what ever limit the POCO has set as a base before charging a different rate. When POCO guy mentioned 'close to a meltdown' that could mean anything. Most of the time their transformers and distribution equipment are sized smaller than what we would size them with using the NEC. They are normally betting on the fact that it will not be as heavily loaded as NEC calculation results, they also have free air around a lot of equipment for cooling it, and if their equipment is overheated it is not inside a building where it is a risk of fire to the building.
For a seasonal load like this they probably do have equipment that is somewhat undersized and assuming it can take the load for a certain number of hours and will have an opportunity to cool down before it is loaded that way again, plus it will only happen for a limited amount of time each year.
Proper overcurrent protection should protect all of the equipment in or on the building from this possible 'meltdown'
Why 3 separate services (from the POCO) They likely would still want to supply this with one transformer and one meter to save some of their costs. Whether or not you use a single main or 3 mains is more up to you or whoever designs it.
Problem with a seasonal load like that is it has to be designed to handle the load when it does run but the rest of the time there is the investment of equipment that is basically idle.