Why would an air conditioning unit specify a maximum breaker size?
The unit won't be protected properly by a larger breaker
While hermetic compressors and fan motors are protected against overload by internal means, or by their drive electronics if you have an inverter-based system, the wiring of the unit still needs to be protected in case it shorts out internally.  As a result of this and the fact that air conditioners universally are complex, multimotor loads, UL has taken pity on installers and inspectors and requires the manufacturer to do the breaker sizing math for us.  (Otherwise, you'd have to dig into Articles 430 and 440 of the NEC and size the breaker yourself, which is a bit fiddly for a multimotor load.)
So, simply use the 15A breaker the manufacturer calls for and call it a day, although I doubt you need 8AWG wire for a unit that draws less than 10A at 240VAC.  In fact, for that run, 14AWG is perfectly fine.  (The chart you found is probably for low-voltage DC circuits, where voltage drop is a far greater limiting factor than at 240V.)  However, there is some merit to future-proofing an air conditioner circuit, and to that end, I'd run ½&quot; ENT as that'll get you up to 50A with THHN wire and the assumption that your panel supports 75°C terminations, then pull the 3 14AWG wires needed through it.