
Adding a cooling unit is impractical in an uninsulated cabinet. You cant just slap on a Peltier cooler and expect results. You would need to add several units depending on the size of your cabinet. Also, you would need heat sinks, temp controllers and probably some other hardware. You may want to talk to Bob Staebell at Aristocrat Humidors for some advice.
Also, Peltier units cool based upon ambient temperatures. They only cool from 5-7 degrees below ambient in the best case.