The device you state is the correct one but it doesn't work 'on its own' - it requires controlling circuitry.
I thought I could just expose the PTC to the inside of the cooler and it would let current through above 40f.
What is the "controlling circuitry" doing?
To digress, your icon says Scotland, do you know any Murrays, or maybe they all moved to Ireland 200 years ago.
My Grandmother was a Murray. For a long time I thought I was part Irish, but I recently found out that something happened in history that moved some of the Murrays to Ireland. But, I do not know what happened.