Most generator engines are pretty unsophisticated compared to modern auto engines ( both gas and diesel ). I think that is part of the problem - they use more fuel than necessary because the design is 40 years out of date. A nice small bike engine of suitable power, coupled to modern BLDC motor and drive in regen mode, and stripped of everything unnecessary would make a good starting point weight wise.
Another good option would be something like a 1 litre 3 cylinder turbo-diesel driving a reasonable (25kw+) generator, on a trailer. That should give good economy, but towing a trailer has its own penalties, so you would not see all the gains from the more economical engine.
My personal opinion is that if you run a small BEV, buy a normal car for longer distances. If you can afford to convert a larger vehicle or a small truck, you might have some space for a generator. It would then be a space/weight trade-off between required minimum range, battery weight, generator weight and power and your budget. I think this is a difficult balancing act, and most is liable to get it wrong.
Dawid