Yep, that is the issue with plastic totes. My suggestion is getting a UTH and putting it on a thermostat ( this one set to 90). Because you having issues with ambient temps too, your going to want to get another UTH and put it on 80, that's your cool side. Another solution is to cut a hole into the lid, then putting mesh, which you can get at home depot, and glueing it to the inside of the lid so the lamp can sit on it. Any heqtspurce you have should have a thermostat. Luckily there are ones online you can get for 20 dollars. Both of mine from Amazon.
Btw binder clips are an amazing choice for clipping the sides of the bin.