Greetings & Welcome!
I think you’re on the right track, last summer I used an experimental unit built by a friend using Peltier, South of Miami. I works well for A/C, & heat down to a little below freezing up here in Minneapolis. Mine seems to cool much quicker than it will heat… In all fairness though, it is cold up here!
Mine has no thermostat, it’s major drawback, and slightly different working principle. It uses a radiator for a heat exchanger with antifreeze in a closed loop using a low power aquarium pump to circulate the water, and mine is powered by a liquid candle, to generate the needed power from peltier chips as well, so no power source is needed at all, and it requires no batteries. It will run for days on just a few ounces of vegetable oil.
For many years I had been using the indirect type of swamp cooler successfully in high humidity areas too though. They also use a radiator for the heat exchanger, and exhaust the moist air back outside. Very effective, and also dehumidifies, and could also be adapted to be candle powered. While more effective than my Peltier powered unit, not having to refill with water regularly is a huge plus.
I am really impressed by the Peltier chips, since they can produce hot, cold, & power. The newer woodstove fans are powered by a Peltier chip instead of the older design which used a heat powered Stirling engine design. The newish Luminiser lanterns, and few other models also use either liquid candles or tea light candles to create bright lanterns which require no batteries.
Good luck with your project & keep us posted!
“Everything should be made as simple as possible." ~ Einstein