> For example, cutting off the system's cooling or ramping it up to its maximum level both resulted in a core temperature change of just 15 degrees or less.
You can't extract energy from a heat engine without cooling. Useful work is done in the process of moving thermal energy from a high temperature location to a low temperature location.
Stirling engines drive because of a temperature differential. If you don't cool then those two temperatures will tend to equalize, becoming less efficient as the entire device warms up.
Why use cooling at all then?