Why not take it a step further and add a built-in battery and auto starter to an inverter generator? Now you have continuous "keep alive" power to your heating system, refrigerator and lighting circuits, and the engine kicks in when the load increases.
The main problem here is the size and weight of the battery and the added cost. Most small 2KVA inverter portable generators are weighing in at around 25Kg dry. Adding a 1Kwhr battery capacity using an SLA battery will take the weight of the hybrid to nearly 100Kg. Suddenly these generators aren't portable anymore. Going to LiFeYPo4 battery will take the weight to around 45kg.
The cost of ownership would also more than double to take efficiency from 20% to 40%. i.e. Hybrid generators would unlikely to cost less than $2000. A better solution would be to have a portable solar powered battery/inverter system module which can be boosted/recharged very quickly when required via mains or generator when required.
Having an electric start like the Yamaha gennies combined with an integrated control signal to boost/recharge the solar battery module when required would be a better solution.
You could then have the choice of purchasing the solar/battery module as an additional module for the Generator to turn it into a hybrid.