Don't even bother with the Free Standing Units that vent out of a window. They are an enormous waste of money and will not help in your situation one bit. There was one that came with my house on the lanai. A total waste.
A Mini Split is probably the only legit shot you have.
I just test fitted a window unit. 10,000 BTU'S. It worked fine but looks like a sore thumb. It's for an emergency strictly.
Just my opinion. Best to do whatever you are going to do with a permit and approval instead of having to fix it when you go to sell. Good Luck.
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