Those mini coolers on amazon uses evaporative cooling or pouring Ice inside of it and aren't worth the price of the box. I don't think I've seen anything smaller than 5K btu which uses a traditional A/C cycle but I;ll suggest an option.

Look up Peltier cooler or Thermoelectric Peltier cooling. To name a few applications the pelitier cooling is used in powerful computers, coolers and beverage can coolers for the car. Its a solid state heat pump which transfers heat from one side of the device to the other and can be purchased in your price range that may do the job. You will need to wire a thermostat across on of the power wires but they run on 12 volts so are low risk of electrocution. An A/C to D/C converter transformer which can handle the wattage requirement is also required. Changing the polarity will turn the cooling plate into a heat plate too.

I have a Coleman cooler 1.3 cubic feet in size with a Peliter cooling plate and works very good keeping items cold. I think the plate is 4" x 4" and pulls 60 watts at 12volts.

We often talk about conditioning living areas, but we really condition volumes. If the 5 SF area is open to another area of your home which is maintained at different conditions, a Peliter-type cooler will not likely do the job. Those are designed for cooling enclosures. You'd need to isolate the volume from the rest of your home in which case your plant would need to live in the dark or you'll need to deal with the additional heat load from a lamp.

Once upon a once I bought a stand-alone AC unit for a bedroom. It was sort of a 2 ' tower-looking thing and you had to empty the water or have somewhere for it to drain. I remember it doing a good job of keeping a small room cool. I bought it at Costco, but they are available in more place now. Your problem will be the cost.

Kevin,

Given how dumb some of the "professionals" are here on houzz.com (with particular emphasis on the HVAC trade) I'd err on the side of too much hand holding rather than too little. A Peltier cooling system would be a great, low-cost solution if a plant didn't need air or light and the volume around the plant could be isolated from the rest of the living area within the OP's budget. The OP might be able to rig up a terrarium like set up with acrylic panels, but that's likely to blow the entire budget without any cooling. A spot cooling system would work, too, but they're going to be higher capacity and higher cost than the OP's budget. A home-made mini-size swamp cooler would be a better option.

Ummm..... maybe I'm missing part of this thread, but where does OP ever say that this plant would be: isolated; no air source; and, with no available light. This could be just a little greenhouse that needs cooling. OP never says.