Yes, this would be more efficient. Because the vast majority of the waste heat created by the relatively inefficient generator would be used for heating the ho

Yes, this would be more efficient. Because the vast majority of the waste heat created by the relatively inefficient generator would be used for heating the home, the electricity produced is nearly free. If you run the heat pump compressor directly from the engine, even more so. The main caveat is that the generator (which would need to be a well insulated, liquid cooled type) would be expensive and would require rebuilding every 10000 hours or so. The efficiency of a small generator is about 1/2 of that of a power plant, but between harvesting 90percent of all waste heat and having no distribution losses, the fuel consumption cost would typically end up lower than the electricity cost of an all electric system. This co-generation design is actually pretty common here fuel oil is the main heating fuel, but it should work even better on gas.