The glue will be too flexible, especially for places where you need the structure to be very stiff. As you noted, the glue flexes and the more it flexes, the more flexible it gets until it fails.
Polyester and epoxy resins flex but not as much as wood glue AND they are much more resistant to failure from repeated small amounts of flexing. The resins made for reinforcing fiber materials have been formulated specifically for uses like your vehicle.
The only way I can see the glue working is if you build a metal frame connecting all the suspension and drivetrain mounting points so the foam coated with glue and glass won't have to support all the heavy bits that are constantly in motion at various frequencies. The body can then mainly support itself and low stress or very low vibration loads like the passengers and the batteries.
The batteries could be a problem. How does the glue, glass and foam composite stand up to a heavy load just sitting on it for a long period of time? Compound shapes will be stronger than a simple, flat "plank" shape. Of course the bottom of the battery holding areas has to be flat. A copy of a section of the battery compartment, finished how it'll be in the completed vehicle, would be a good test article to load up with batteries or weighted boxes the size of batteries.
Let sit for a period of time, in a spot where it will experience a variety of environmental conditions, especially hot temperatures. If the bottom doesn't deform then you've a design that can at least withstand long term static loading.
PVA glue loses strength at elevated temperature. I've tried to find a graph for the Titebond line but so far have only found what happens at 150F. Titebond III loses half its strength at that temperature, so I'd assume it has a pretty steep drop as temperature climbs. What happens to it at 100F? How hot will the motor get?
Another issue with PVA is bonding of it to dried PVA.
Differences Between Titebond Glues | The Wood Whisperer
If you don't want to spray a coat of epoxy on the foam to protect it from the polyester resin, you can roll and brush it. A skim coat of an epoxy based body filler would give the foam a smoother surface, reducing the amount of epoxy resin required.
West System tested polyester gel coat over epoxy and found that when the epoxy was properly cured and the surface prepared, the polyester formed a very strong bond to the epoxy.
WEST SYSTEM | Projects | Fiberglass Boat Repair and Restoration - Polyester over epoxy
The better you do the finish work on this body, the easier it will be to use it as a plug to make molds. Using polyester resin for the molds on a plug coated in wood glue that loses 50% strength at 150F could be a disaster since polyester resin can get pretty hot while curing.