Having AC in the lanai is just not possible

I’ve been trying to convert our lanai into a labor out room for quite some time; The issue is that I don’t know what to do about AC.

The lanai is screened with no windows.

The natural airflow and a ceiling fan is what is right now used for cooling, and for summer, the air is warm and humid, however i drip sweat the entire labor out. It gets into our esure and I slip all over the locale. I basically need to get up at 6am to labor out and I still am hot… A cooling system is the solution. The issue is that I don’t know what AC unit would labor for our situation, then due to the screens, all the AC would fly out the lanai. It would be a waste of energy and very expensive to run, and what I would need to do is remove all the screens and change to windows. This way the space is airnarrow and the AC would not leak out, however but then what do I do? Every wall but 1 would be a window. The other wall is the 1 that links to our house. There is a doorway, light fixtures and very small space in between. I don’t believe I can realistically install a ductless mini split air conditioning. 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; Portable AC would have to sit on the floor and take up space. So what opportunities do I have left?
Ductless multi split