If you are ever feeling like burning down your apartment ;), the route I'd take is buying a 5-20p extension cable, cutting it open, and splicing in a 120V switch into the middle of it. You can find a few smart switches that fit that bill for whatever relay tech you are interested. It's non-invasive for the AC unit and can be quickly removed.
There's only 3 wires to deal with as well so it's not THAT scary to make your own cable (but I'd understand why you might feel differently).
First, I'm not referring to a smart plug. I'm referring to a thermostat-controlled plug. You were afraid that it would rapidly turn the AC on and off, which would burn out the motor.
They don't do that. Mine (when I use them with a heater) run it for a few minutes, then turn it off for awhile. They don't rapidly turn the heater on and off. I assume it's the same way when run in cooling mode.
The people who make them aren't stupid.
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WRT a rely: They are safe, you just need to look at the rating for the load. (The point of a relay is to switch a high current / voltage load.) I use an industrial one for an outdoor outlet: It can handle the full 15 amps of the circuit safely. (I use it to adapt a smart switch that can only handle light loads, including a relay.)
> It can handle the full 15 amps of the circuit safely.
OP would need a more beefy relay for an AC unit. 20A minimum, but honestly a 30 or 40A relay would be warranted as AC units have a pretty large "spike" when they turn on which generally exceeds the amp rating of their circuit. It's short enough to not trigger a circuit breaker, but it's something to be aware of when designing things like relays.
I'm fairly certain those exist. The "click" that a car charger makes is a commuter (sp?) (basically a relay) connecting the car to voltage. Mine handles 40 amps.
There's only 3 wires to deal with as well so it's not THAT scary to make your own cable (but I'd understand why you might feel differently).