Since nobody seems to be entertaining your idea: Sure. Maybe. The specifics aren't really worked out because nobody has bothered to work them out. Maybe it's

Since nobody seems to be entertaining your idea: Sure. Maybe. The specifics aren't really worked out because nobody has bothered to work them out. Maybe it's better, maybe not. If you manage to find a way to extract 99% of the waste heat of the engine without choking it, you're probably still not getting 99.9% efficiency out of it, because ICEs do not always have stocheometric air/fuel mixes, and generally have a lot of inefficiencies in converting all of the energy present in the fuel, especially a small 2-stroke engine as seen in home generators. On top of that, you would need to devise a way to run a heat pump compressor off of an ICE, which is probably possible but you'd have to make that. So if you're a company looking to make a product to disrupt the electric heat pump industry, you would have to solve all of these problems, and at the end of the day you'd be barely more efficient than burning fuel in an electric plant and transmitting the power that way. Since current costs to the consumer are ballooning, you may edge out electricity in costs, but I highly doubt you're gaining that much efficiency because there isn't much to be had (modern natural gas electric plants are unfathomably efficient)