If you still have shortage of electricity, then consider running the genset, use the power to run the heatpump, and collect the waste heat of the genset, for >100% total efficiency.
Probably the easiest way to use part of the waste heat is to direct the warmed cooling air from the generator through the outdoor airflow of the heat pump so that is is using somewhat warm air as a heat source instead of plain cold outdoor air. This will raise the coefficient of performance of the heat pump for sure.
Not by much - what you propose is a variation of recurring modification theme, but people fail to understand these outdoor units process just massive amounts of air through them. This is because air has poor thermal capacity, so a lot of flow is needed. Whether you mix solar-heated air (one common modification idea), warm ventilation exhaust air from the building (another), or exhaust gas of the genset (yours), it's going to mix with 10-100 times larger volume of outdoor air, so the benefit is almost negligible, maybe one-two degrees C and 0.1 units in COP.
The next step is to suggest casing the whole outdoor unit to prevent this airflow, but this is trying to break the laws of physics: the only reason why seemingly overunity efficiency is possible with heatpump, is they actually harvest thermal energy in our atmosphere, which is basically an infinite source of energy. Try to modify them to harvest some limited resource (like waste heat source), and you are going to be limited to maybe 50% efficiency, not 300% some might expect.
Probably the soot and other crap in the exhaust gas does more damage to the performance of the evaporator than it helps.
But, something I would seriously consider, if you truly need a backup system off-grid, or for backup during blackouts, or just to help with peak expensive hours of electricity, is getting a
decent genset (not a cheap toy which breaks after 10 hours of use), which is water-cooled, so you can trivially just connect it to your hydronic heating circuit through a heat exchanger. Then run bog-standard air-to-air heatpump - which you should have anyway - with that electricity, of course along with all your normal electric loads. This way you can tap into most of the waste heat to heat up the radiators, some in rooms and hallways etc. that are outside the range of the air-to-air heatpump; while the air-to-air heatpump heats the living room, maybe.
If you run 30% efficient genset to run a heatpump at 300% COP, and harvest half of waste heat of the generator, you will have 125% efficiency from gasoline chemical energy to heat. Compare this to running direct electric heating with the same genset plus harvest half of the waste heat = 65%; or direct electric heating with no waste heat harvest = 30%.
But practically, just do the right thing, get a heatpump or two (air-to-air units are cheap and provide summertime cooling), get some PV, and have your neighbors do the same. Then you can forget about the whole generator idea.