As usual, I’m probably going to add to the confusion, but here it goes…
You mentioned heat pump, so I’m thinking a central heating/cooling system…I’m not familiar with the phrase “heat strip”, other than electric baseboard heat. My lack of knowledge here may not be important. If you do have central heat and air, do you also have a central humidifier built in it? These types of units can be problematic, and may lead to symptoms like you are describing. In fact, many contractors would avoid installing them in heat pump applications where the temperature is usually much cooler than that of forced-air heaters using gas or oil.
I can offer a more personal experience. Upon moving into my home last June, I was faced with a similar smell when the ac started. My inspections led me to the whole-house humidifier, which inside, looked like a biology experiment gone crazy. I removed it, and the smell went away. Oh, yes, at the same time we also had the ducts power cleaned, a recommendation made to me during my study of the problem, primarily due to my wife’s allergies and the fact that the previous owner had cats. The combination improved our home’s air quality considerably.
My study of the problem also uncovered other recommendations for the moldy odor, including a clogged or slow-draining evaporator tube, or even a clogged air filter, especially the hi-tech ones.
Of course, none of this relates to your crawlspace, and maybe that’s accurate. If your crawlspace does not have the odor you detect inside your home, then the two are probably not related. Ideally, the ductwork for your central system should be isolated from the environment in your crawlspace anyway, but that is not always true.
Even so, I would not rush out and fire your exterminator. Although not naming what he found, he did point it out to you, and regardless of whether or not it is causing your smelly situation, your crawlspace should nonetheless be examined by a professional to make sure there isn’t another problem there.