Wonder if you can just remove the dead battery and fool the electronics into thinking the battery's there. Then you have the smug satisfaction of operating a "hybrid" vehicle with the convenience of the gas engine.
You would have to know a lot about how the computer(s) that run the car work and how to program them. It would be along the lines of reprogramming the basic input output system in your desktop or laptop only more complicated with less publicly available software “tools” available and a much smaller base of individuals with this type of knowledge.
“Hybrids” are very complicated beasts, much more complicated than either electric or hydrocarbon fueled vehicles. This is especially true of the transmissions which have to take in energy from both the engine and the motor/generator. It would be more of a challenge than anyone who I know could take on... and I know some very adept people.
The actual way to do it would be to get another non-hybrid version of the same car - that had been totaled and try to transplant the engine, transmission, electronics, and other items into your dead hybrid.