Mini splits are incredibly cheap - well under $1000.
The AC companies routinely charge 400% of the price PLUS installation costs. Min Splits are also very simple to install. But, this is TV, so go figure
Here's where I've always bought my AC stuff.. this particular model is the smallest but still massive overkill for a lanai.
🔥 9k BTU 18 SEER MrCool Advantage Ductless Heat Pump Split System 3rd Generation - 230v - HA18427 • Ingrams Water & Air