You could put a small compressor outside and connect a well-insulated hose to pump the refrigerant inside, just like a split system air conditioner. Then the noise in your bedroom will be no worse than the occasional hum of a fridge in the other room.
As someone who lives in a humid climate, I wouldn't even think of using an evaporative cooler anywhere in my home, for any purpose, period. It's either compressors or nothing.
I believe the Eight sleep uses a Peltier cooler. Not terribly efficient, but the ΔT and the load are low (a human body puts out <100W while asleep). Plus, a Peltier can also heat with some supporting electronics.
I would think these would be too loud for a comfortable sleep environment.