While the suggestions are well and good the criteria that was set aside by OP was not addressed completely.
Moving a heavy equipment requires absolute care to avoid damage and injury to personnel and equipment. Hence, there are specialists called riggers.
The moving part (ie.) loading, unloading, lifting and spotting the load in place are straightforward.
OP mentioned that this equipment is going to be hauled-in (moved) into an elevator.
Most commercial medium sized passenger elevators are rated 2500 lbs to 5000 lbs
Class A freight elevator is rated almost the same as a passenger elevator.
OP did not state the rated capacity of the elevator that he intends to use. . . although he stated
“it will fit and the weight capacity is OK”.
The recommended loading for elevators is limited to 25% of their rated load.
So, an equipment weighing 3200 lbs would require a rated 12800 lbs. elevator . It would be a non-standard, custom made elevator.
Don’t be fooled with elevator rating thinking your 3200 lbs load can be handled by a 5000 lbs elevator.
I had hoisted an MCC /Distribution panel to a rooftop, inside a shed and I had to hire a Sikorsky Helicopter Crane to lift it to the roof and roll it in place.
The building elevator wasn’t big enough-- beside the openings were just regular door openings.
It cost the company over $6500 for a forty five minute gig. This was twenty years ago.
The helicopter was based at an airport about 3 miles from the company grounds.
If you haven’t done this before check below link:
Better still, as one poster said, hire a rigger.
If still in doubt check with your OSHA Checklist. Don't hold me for something I missed.
https://www.elevators.com/freight-elevator-classes/