There isn't anything new really about this idea. Besides the assertion made at the top that the question is asking for a Plug-in Hybrid, there are examples where folks have built a generator to charge an EV while driving.
The AC Propulsion Long Ranger could charge the tZero while the car was being driven. The tZero was the car which proved to the founders of Tesla Motors that they could build what became the Tesla Roadster. It was a high performance electric car that initially had a lead-acid battery pack, and later a lithium-ion battery pack. The Long Ranger trailer had a 500cc Kawasaki engine, a 40 liter gas tank, produced 20 kiloWatts of charging power, and achieved 30-35 MPG in actual usage.
AC Propulsion did not make the only implementation of this idea. The idea has come up many many times in the past. One of the more famous is a trailer developed by Ingineer, a.k.a. Phil, who put a Capstone Microturbine on a trailer and connected that to a Nissan Leaf. The generator connected to the Leaf through the CHAdeMO port but it's not clear whether the setup allowed the car to be charged while it was being driven. It produced 30 kW of charging power and ran on propane.
The simple DIY implementation is to put a trailer hitch on the car, then haul a trailer that has a generator on it. If the generator charges the car through its J1772 port, then it need only be sized big enough for the 6 kiloWatts or so that on-board chargers use. On the other hand charging the electric car from the generator through the fast charging port requires a much larger generator. In the charging-while-driving scenario, the system would detect a low state of charge and kick the generator into action - perhaps after getting confirmation from the driver.
Business model for occasional use towable generators for electric cars
There's a fairly obvious business model here. Namely - rent charging trailers at the outer edges of urban areas. Since the areas between metropolitan areas have few charging stations, this is where EV drivers need the most assistance. The charging trailer would, in other words, be most useful for driving through areas with few charging stations.
Folks rent all kinds of things for driving trips, this would be no different.
Such a trailer isn't useful in most urban areas since the charging networks in cities are usually sufficient.
As the inter-city charging networks get built, there will be less need for such a trailer. If someone can just stop at a fast charging station, why do they need to rent a generator?
Technical issues against using a towable generator on an electric car
The idea of a towable gasoline or diesel powered generator for an electric car is sound, and there is a clear business model for it. However, there are at least two problems with the idea.
Emissions regulations The first is that emissions rules for different classes of cars are based on the manufacturing. Various kinds of government incentives are awarded based on the emissions ratings.
Before you dismiss that concern as nonsense consider the Dieselgate problem that was discovered at Volkswagen and several other automakers. In that case it was discovered several vehicles, primarily with turbo-charged diesel engines, were nowhere near as clean as was promised by the manufacturer. Volkswagen promised great emissions for their TDI Diesel cars, and had lab tests to back them up. The problem was that Volkswagen designed the car to detect when it was undergoing an emissions test, so the car engine would behave one way (producing excellent emissions results), while the car engine would operate in its normal way that produced lots of NOX pollution. Volkswagen was not the only automaker, and all involved have had to pay out huge fines and some have gone to jail. These individuals and companies were charged with fraud and other crimes.
A battery EV powered by a towed gasoline powered generator is clearly not operating as a zero emissions vehicle.
Electric cars are designed to not charge while driving A safety feature built into electric car charging protocols causes the drive train to be locked out while the car is plugged in and charging. Obviously in the normal case when an electric car is plugged-in to a charging station, it should not be driven. Gasoline car drivers accidentally do the "drive-off" thing from time to time causing various amounts of havoc. Automakers obviously wanted to avoid the drive-off problem with electric vehicles, and designed that into the charging protocol.
It's a good safety feature that would prevent the simple implementation of a charging trailer. Namely, a charging cord would be run from the trailer to the charging port and left plugged in while driving.
That's a nice simple scenario isn't it? Problem is this safety system will prevent that from occurring.
Hence the DIY charging trailer for electric cars would not be able to use the simple solution. Given the awesome hacks that some have done to their electric cars, maybe someone could rig a direct-to-the-battery-pack connection that integrated with the on-board battery management and charging management, in a way that bypassed the safety system. Or maybe not.
Clearly a manufacturer could design such a feature into their car. It would possibly use a special charging port that bypassed the safety system. Or perhaps the charging trailer would identify itself to the car through the regular charging port - through a protocol that does not exist today - allowing the regular charging port to be used.
As a bonus let's consider a third issue: Are charging ports able to securely hold the charging cord while the car is driving? Consider a charging cord plugged into a charging port. The manufacturer obviously assumed this port would only be used while the car is sitting still at a charging station. Since the charging port would not be used while the car is driving, it's possible for driving vibrations to cause the cord to fall out of the port.
Some charging ports lock the charging cord while the car is charging. This lock might be secure enough to hold the cord while driving, but probably not.
Avoiding driving-while-charging problems with charging trailer for electric cars
In the previous section we talked about the technical infeasibility of the idea. Now lets talk about a possible solution.
Charging while car is stationary Obviously the theoretical DIY charging trailer for an electric car could be used while the driver is lounging at a rest stop. The interlock preventing an electric car from driving while charging does not care whether the charge current is coming from a generator or a regular charging station.
The problem here is avoiding a multi-hour wait for charging to finish. At 6 kiloWatts, which is the typical charge rate for level 2 AC charging, a 4 hour wait is required to charge a 100-ish mile range electric car, and an 8 hour wait is required for a 200-ish mile range BEV. Who is going to have 4-8 hours of fuel to power their generator? Who is willing to wait that long for the generator to charge the car?
In other words it will be desirable for the generator to support fast charging. But, generators capable of 20 kiloWatts or 50 kiloWatts or 100 kiloWatts are large. And expensive.