BC52,
Is your HEI a GM style. This style starts and runs on full voltage, so whatever wiring method is using a START/RUN circuit that seems to be failing, isn't necessary. I'm not sure what this would be...either the keyswitch or solenoid relay, but it should/could be jumped around or rewired to negate it.
I wonder if the ignition module is going bad and won't function properly when the starter is drawing the battery down a bit. I'm out of my league...maybe someone else will come along with the answer.
Did it just start doing this?
Stuart