Could it be the heat pad? May I suggest before breaking everything, try a 50W reptile heat lamp, as an experiment. That's what my PID is using, in a bigger cubic area than yours, and it will bring the temp up to 95 no worries even if the outside temp is 40.
I am interested in what you are doing Phoebee, because I would prefer to use a heat pad also, for the reasons you stated. But my technical guy didn't think it would get the temp up but the science he threw at me was beyond me.
Will follow your results with interest.