Hate to be a downer but I wouln't risk it, its not worth risking burning down your house or dieing of smoke inhalation or toxic fumes.
A light bulb would be on an insulated ceramic light base, and people use thermostats, so its less likely to start a fire, however precautions are important.
Wireing up a high temp heating wire, its not got the insulated base, the heats going to transfer, and those are going to be some really nasty fumes. What could burn? the cooler is foam or plastic right? if it overheats and heats what ts attached melt and burn the walls of the cooler.