[SS427]

It is likely the solenoid and is very common especially with headers. The solenoid gets so hot that it builds up so much resistance and has huge voltage drops so that there is not enough left to pull the solenoid in. Normally a new solenoid will solve the issue as will a solenoid heat shield that are readily available reproduction. You said the battery cables are new otherwise that could also be a cause and for the same reason, they loose their ability to carry enough current as the resistance is too high.

Back in my drag racing days we solved the repeating issue by wiring in a Ford solenoid and keeping it out of the hottest part of the engine. Problem solved but likely not one you would use on a restored car.