Having A/C in the lanai is just not possible

There is a doorway, light fixtures plus very small space in between.

I’ve been trying to convert my lanai into a work out room for quite some time, the issue is that I don’t think what to do about AC. The lanai is screened with no windows. The natural airflow plus a ceiling fan is what is right now used for cooling; For summer, the air is sizzling plus humid, then i drip sweat the entire work out. It gets into my eyes plus I slip all over the locale. I basically need to get up at 6am to work out plus I still am warm; A cooling system is the solution. The issue is that I don’t think what A/C component would work for my situation, however due to the screens, all the A/C would fly out the lanai. It would be a waste of energy plus very expensive to run; What I would need to do is remove all the screens plus change to windows. This way the space is airtight plus the A/C would not leak out, then but then what do I do? Every wall however a single would be a window. The other wall is the a single that links to my house. There is a doorway, light fixtures plus very small space in between. I don’t think I can realistically install a ductless mini split I need to mount the indoor air handler on an exterior wall. That is not possible. I can’t exactly link ductwork to the lanai either, but portable A/C would have to rest on the floor plus take up space, but so what options do I have left?

Wireless thermostat