We've used dish soap in our black tank a time or two and water softener too.
Drove some time before filling completely and flushing. I'm not sure we needed to but there is no buildup in the tank and sensors in all tanks work fine. The gray tank gets daily soap and shampoo, and dish soap and laundry detergent and softener. No grease except minute trace amounts. After we flush tanks, the gray as last the hose smell like a cologne. So soap certainly hasn't hurt them.
i wouldn't be putting grease in the holding tanks and certainly not the black tank.The water softener we add and black tank additives seem enough. Probably the softener and soap is wasted but done when we sit a while not emptying.
We don't leave the black tank nor gray tank valves open after seeing something crawl out of the sewer systems. And witnessing what happens with prymids, when sitting for a long time or short time, with no water in the tank.
If your sensor and not working well the short of it is put some soap and softener in it and mostly fill the tank and drive a bumpy road. Won't hurt.