I do own a Yamaha 2800i. I've had it almost 3 yrs with NO issues whatsoever.
I'm not sure on a 15K A/C unit. It will pull 7500 (NOT a misprint) watts for 3 seconds, so I know it has the starting capacity. If the compressor doesn't run for more than 20 min at a time, the little generator will pull 2800watts/23amps so I think it will carry it. To be sure check the power requirements for the 15K unit and check it against those numbers. I'm sure that you're load capacity for other items would be limited (no coffee makers, microwave, other high draw items) for SIMULTANEOUS use. But if you plan your usage, you should be able to put the A/C in the fan only mode while using the other items and then go back to the A/C mode when you're finished with the other items.
I did check one website and it listed the power draw for the 15K A/C as 15.4 amps which is well within the limits, including starting, of the Yamaha 2800i generator. It would give you about 7amps to spare when the compressor is running, enough to run a decent sized TV.
One other thing; you might want to consider putting your fridge on propane while running the A/C on this generator to ensure the total draw is within limits.