Since the OP asked about natural gas heaters I'll continue answering that question. There are permanently (wall) mounted un-vented heaters rated for residential heat. They have built in safety devices which keep you from dieing of obnoxious fumes.
At other homes I've owned them and installed them. If that is what you want I can tell you more. Spoiler alert I would not install one here, in the lanai or anywhere else. They have a place and purpose and but it is not here.
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