Hi (Customer) ... Unfortunate situation to say the least. Here's a method we used to use several years ago here in Iowa to dig footings for homes in the dead of winter. Worked surprisingly well and I believe it will work for you. Corn cobs, straw and kerosene. We piled this along the line of the intended footing and let it smolder several hours. I recall doing this one time when the temperature was 8 below zero. Your problem is most likely the waste line pipe from the house to the tank and not the tank itself. The chemicals and heat generated in the tank should not allow freezing. If you can't find corn cobs look for something similar. Old oil, old tires, dense tree limbs, etc. Anything that will smolder and not burn out too quickly.
NOTE: Hold off on making the "acceptance" if you have additional questions for me.
Hello (Customer)... Corn cobs??? You bet.. We got plenty of them. I live in Iowa. With out the ability to have an initial fire to get some sort of debris smoldering I suppose some other approach will be needed. It was the intention of the kerosene to get the straw and cobs "glowing" (cooking as you say) which happens just after lighting it on fire. Then another question comes to mind. Even if you were able to unthaw this pipe what will keep it from just freezing up again? Have you asked anyone that lives in a similar location what they do with their septic system in the winter?
I'm going to send this message back to you as in "information request" which will in effect keep your question open for all of the other experts to see. Perhaps someone will have a "creative" thought other than a fire.
Sounds like you may be on the road to recovery with your solutions. When you use the pressure washer if there is any way to get hot water to it would be a great help.