Mini split system to heat and cool a gymnasium?

You should be dealing with an architect and commercial grade contractors for this project.   Like your building system is determined first and from that your a/c system is specified. Do you have sufficient power and where is it?  Lighting? Water? What about bathrooms? Laundry?  How are they air conditioned?

because of your ceiling height, think i'd want double the normally recommended capacity ... your volume to be heated and cooled is double or more 'normal'.  and I'd look carefully into the assumption about insulation .. your gym may well have a lot less than 'normal' ... which makes a huge difference.  [so much so that installing insulation in the ceiling when it is being built is probably a big money saver.]  -- grampa

it can be done if its sized right.  we had a old warehouse building that a customer turned into a dance studio as he grew he turned different floors into studios.  The top 2 floors was open floor from the front of the building to back probably 40 ft by 100.  Those floors we did with minisplit. Nice thing is that most units now don't only heat an cool but can be used for humidity control when cooling is needed.   Which came in handy an would probably be useful in your gym.