If you used a cooler you would have to monitor the temperature and be religious about changing/adding ice to keep the temperature down around 37°. I personally wouldn't do it.
The question becomes would you use a refrigerator on an ongoing basis for more curing in the future? If so, then yes it makes sense to get one.
Three years ago my wife got tired of my sausages, hams, etc. taking up space in the fridge and made me get my own. I picked up a used one for $50 in great shape and now I can't imagine not having one just for curing.
Think about it this way: It would only take your pork bellies spoiling one time to justify the cost of buying a used refrigerator.
Hope this helps.
Cliff