Great idea. I am trying the same thing with an old water cooler we used when I coached my daughter's track team. It already has a hole in the bottom that fits my old heating pad cord, so no drilling (I did have to clip the edges of the cord to fit the narrow hole. I also used an easy stick hook to hang the cord on the inside of the cooler. If it works, I will fill the hole with bathtub caulking.
The big win is that most of the $150 warming baskets are much smaller than my water cooler. I just hope the heating pad produces enough heat to warm everything. Cost for me was $0.00 if all goes well.