What most people do is run the heat tape along the back of the cooler. Helps with heat regulation. If you put it on the bottom under all that stuff you are going to have a hard time keeping the temperature uniform and if you put the egg boxes on top of the heat tape you're going to have a much higher ambient temp in the bottom boxes than the ones above it.
What I did was run a foot and a half in the back of my wine fridge incubator. Installed a computer fan on the top to push the warm air down and circulate the temperature. And I placed water bottles on the bottom. Then stacked my egg boxes on the water bottles.
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