"How would I check the accelerator pump? It does restart so I assume the ignition is ok"
No.1 - Never a$$-u-me anything. I thought marriage would be a good idea and got tricked on that one.
To check the fuel, remove the ACL and operate the throttle linkage while looking for a fuel shot down the throat of the carb. If no shot, most likely an empty fuel bowl due to heat soak/boiling of fuel. You may also have bad ignition parts (condenser) that shuts the car down with heat.
If no fuel, you will have to isolate the carbs with phenolic carb spacers and maybe insulate the fuel line from pump to carb.
Also verify pump pressure/volume delivery and any possible air leaks in the fuel line from tank to pump.
It's usually ethanol blended fuel as at temp it experiences blend separation and is more prone to evaporation due to a lower boiling point.
W (BY GOD) V - EASTERN PANHANDLE