How strong is epoxy? A lot stronger than any PVA glue. Some numbers for West Systems
WEST SYSTEM | Epoxy Resins and Hardeners - Physical Properties
It's not just about yield strength. Also check out the flexural strength, hardness, compression yield and the rest.
I've been looking and have yet to find any such numbers for Titebond III. Can it withstand the loads the suspension and other components will put on it without cold flowing?
Wood glues are designed for wood - fairly rigid and non-compressible. The glue holds the wood parts together. The wood and the shape of the glued joints are responsible for most of the strength. Some amount of flexibility keeps the glue from cracking when put under stress. If a piece of furniture is stressed enough for it to fail, it doesn't matter much whether it's the glue or the wood that breaks.
Resins designed for use with fibers in composites are made to not compress, creep, cold flow, flex too much or be too rigid. Putting it over a foam core, the foam adds strength much the way the gas in a blimp keeps the envelope from buckling. With no empty space inside, it takes quite a bit more stress to buckle a thin composite layer. Without the foam filling, the composite layer would need to be thicker to prevent buckling.
Coating your foam car body with glass fiber and wood glue may work for a while, but I wouldn't expect it to be usable long term. Given the much lower tensile strength and the other critical properties of the PVA that are unknown - that's just too much that's unknown to use a product in such an "off label" application where human safety is involved.
Contact the company that makes the glue and ask if they have any of the data like is on that epoxy chart.
Getting injured or dead because the body of your homebuilt vehicle breaks would really suck.